A solid conducting sphere has net positive charge and radiusR = 0.600 m . At a point 1.20 m from the center of the sphere, the electric potential due to the charge on the sphere is 18.0 V . Assume that V = 0 at an infinite distance from the sphere.

What is the electric potential at the center of the sphere?

Answer : The answer is, 5.97, 24

Explanation :

Scientific notation : It is the representation of expressing the numbers that are too big or too small and are represented in the decimal form with one digit before the decimal point times 10 raise to the power.

For example :

5000 is written as [tex]5.0\times 10^3[/tex]

889.9 is written as [tex]8.899\times 10^{-2}[/tex]

In this examples, 5000 and 889.9 are written in the standard notation and [tex]5.0\times 10^3[/tex]  and [tex]8.899\times 10^{-2}[/tex]  are written in the scientific notation.

If the decimal is shifting to right side, the power of 10 is negative and if the decimal is shifting to left side, the power of 10 is positive.

As we are given the 5,970,000,000,000,000,000,000,000 in standard notation.

Now converting this into scientific notation, we get:

[tex]\Rightarrow 5,970,000,000,000,000,000,000,000=5.97\times 10^{24}[/tex]

As, the decimal point is shifting to left side, thus the power of 10 is positive.

Hence, the answer is, [tex]5.97\times 10^{24}[/tex]

Now the answer is comparing to [tex]m.\times 10^n[/tex]

So, m = 5.97 and n = 24

Thus, the answer is, 5.97, 24

Answer:

D. 4 m

Explanation:

According to the work-energy theorem, the work done by the force acting on a body modifies its kinetic energy.

[tex]W=\Delta K\\W=K_f-K_i\\W=\frac{mv_f^2}{2}-\frac{mv_i^2}{2}[/tex]

The car comes to rest after a given distance, so [tex]v_f=0[/tex]. Recall the definition of work [tex]W=Fdcos\theta[/tex], here F is the constant force acting on the car, d is its traveled distance and [tex]\theta[/tex] is the angle between the force and the displacement, since friction force acts opposite to the direction of motion [tex]\theta=180^\circ[/tex] :

[tex]-Fd=-\frac{mv_i^2}{2}\\d=\frac{mv_i^2}{2F}[/tex]

We have:

[tex]v_i'=1\frac{m}{s}\\v_i=0.5\frac{m}{s}\\v_i'=2vi[/tex]

Thus:

[tex]d'=\frac{2m(v_i')^2}{2F}\\d'=\frac{2m(2v_i)^2}{2F}\\d'=4\frac{2mv_i^2}{2F}\\d'=4d\\d'=4(1m)\\d'=4m[/tex]

Answer:

33.7410805301 s

22336.5953109 m

Explanation:

t = Time taken

u = Initial velocity

v = Final velocity = [tex]4\times 331[/tex]

s = Displacement

a = Acceleration = [tex]4g=4\times 9.81\ m/s^2[/tex]

g = Acceleration due to gravity = 9.81 m/s²

[tex]v=u+at\\\Rightarrow t=\dfrac{v-u}{a}\\\Rightarrow t=\dfrac{4\times 331-0}{4\times 9.81}\\\Rightarrow t=33.7410805301\ s[/tex]

The time taken is 33.7410805301 s

[tex]s=ut+\dfrac{1}{2}at^2\\\Rightarrow s=0\times t+\dfrac{1}{2}\times 4\times 9.81\times 33.7410805301^2\\\Rightarrow s=22336.5953109\ m[/tex]

The plane would travel 22336.5953109 m

The shortest time is approximately 33.74 seconds, and the distance traveled is approximately 22.32 kilometers.

To solve this problem, we will first need to convert Mach 4 to a speed in meters per second, then use the kinematic equations for constant acceleration to find the time and distance.

(a) The speed of sound in cold air is given as 331 m/s. Therefore, Mach 4 is:

[tex]\[ v = 4 \times 331 \text{ m/s} = 1324 \text{ m/s} \][/tex]

The acceleration that the pilot can withstand without graying out is less than 4 g. Since 1 g is equal to 9.81 m/s², the maximum acceleration the pilot can withstand is:

[tex]\[ a_{\text{max}} = 4 \times 9.81 \text{ m/s}^2 = 39.24 \text{ m/s}^2 \][/tex]

Using the kinematic equation that relates initial velocity, final velocity, acceleration, and time:

[tex]\[ v = u + at \]\\ where \( v \) is the final velocity, \( u \) is the initial velocity (0 m/s since the pilot starts from rest), \( a \) is the acceleration, and \( t \) is the time. We can solve for \( t \): \[ 1324 \text{ m/s} = 0 \text{ m/s} + (39.24 \text{ m/s}^2)t \] \[ t = \frac{1324 \text{ m/s}}{39.24 \text{ m/s}^2} \] \[ t \approx 33.74 \text{ s} \][/tex]

(b) To find the distance traveled during this time, we use the kinematic equation:

[tex]\[ s = ut + \frac{1}{2}at^2 \]\\ where \( s \) is the distance, \( u \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time. Since \( u = 0 \) m/s: \[ s = 0 \times t + \frac{1}{2}(39.24 \text{ m/s}^2)t^2 \] \[ s = \frac{1}{2}(39.24 \text{ m/s}^2)(33.74 \text{ s})^2 \] \[ s \approx \frac{1}{2}(39.24 \text{ m/s}^2)(1137.52 \text{ s}^2) \] \[ s \approx 22320.54 \text{ m} \] \[ s \approx 22.32 \text{ km} \][/tex]

Therefore, the shortest time the pilot can take to reach Mach 4 without graying out is approximately 33.74 seconds, and the distance traveled during this period of acceleration is approximately 22.32 kilometers.

Answer:

221.17 kJ

Explanation: Note the heat of vaporization is in kJ/mol,then to determine the number of moles of water: divide the mass by 18. Then multiply the number of moles by the molar heat of vaporization of water.

N = 97.6 ÷ 18

Q=molar heat *moles

Q = (40.79) * (97.6 ÷ 18)

This is approximately 221.17 kJ

Answer:

Explanation:

a ) Height to be cleared = 5 - 1.6 = 3.4 m

Horizontal distance to be cleared = 5 m .

angle of throw = 56°

here y = 3.4 , x = 5 , θ = 56

equation of trajectory

y = x tanθ - 1/2 g ( x/ucosθ)²

3.4 = 5 tan56 - 1/2 g ( 5/ucos56)²

3.4 = 7.4 - 122.5 / .3125u²

122.5 / .3125u² = 4

u² = 98

u = 9.9 m /s

Range = u² sin 2 x 56 / g

= 9.9 x 9.9 x .927 / 9.8

= 9.27 m

horizontal distance be-yond the fence will the rock land on the ground

= 9.27 - 5

= 4.27 m

Answer:

a) [tex]u=13.3032\ m.s^{-1}[/tex]

b) [tex]s=3.7597\ m[/tex]

Explanation:

Given:

horizontal distance between the fence and the point of throwing, [tex]r=14\ m[/tex]

height of the fence from the ground, [tex]h_f=5\ m[/tex]

height of projecting the throw above the ground, [tex]h'=1.6\ m[/tex]

angle of projection of throw from the horizontal, [tex]\theta=56^{\circ}[/tex]

The horizontal component of the velocity that remains constant throughout the motion:

[tex]u_x=u\cos\theta[/tex]

Now the time taken to reach the distance of the fence:

use equation of motion,

[tex]t_f=\frac{r}{u_x}[/tex]

[tex]t_f=\frac{14}{u.\cos56}[/tex] .................................(1)

Now the time taken to reach the fence height (this height must be attained on the event of descending motion of the rock for the velocity to be minimum).

Maximum Height of the projectile:

[tex]v_y^2=u_y^2-2\times g.h_m[/tex]

[tex]h_m=\frac{u_y^2}{19.6}[/tex] ........................(4)

Now the height descended form the maximum height to reach the top of the fence:

[tex]\Delta h=h_m-h'[/tex]

[tex]\Delta h=(\frac{u_y^2}{19.6} -3.4)\ m[/tex]

time taken to descent this height from the top height:

[tex]\Delta h=u_{yt}.t_d+\frac{1}{2} \times g.t_d^2[/tex]

where:

[tex]u_{yt}=[/tex]  initial vertical velocity at the top point

[tex]t_d=[/tex] time of descend

[tex](\frac{u_y^2}{19.6} -3.4)=0+0.5\times 9.8\times t_d^2[/tex]

[tex]t_d=\sqrt{(\frac{u_y^2}{96.04} -\frac{3.4}{4.9} )}[/tex]..............................(2)

So we find the time taken by the rock to reach the top of projectile where the vertical velocity is zero:

[tex]v_y=u_y-g.t_t[/tex]

where:

[tex]u_y=[/tex] initial vertical velocity

[tex]v_y=[/tex] final vertical velocity

[tex]t_t=[/tex] time taken to reach the top height of the projectile

[tex]0=u_y-g.t_t[/tex]

[tex]t_t=\frac{u_y}{9.8}\ seconds[/tex] .................................(3)

Now the combined events of vertical and horizontal direction must take at the same time as the projectile is thrown:

So,

[tex]t_f=t_t+t_d[/tex]

[tex]\frac{14}{u.\cos56}=\frac{u_y}{9.8} +\sqrt{(\frac{u_y^2}{96.04} -\frac{3.4}{4.9} )}[/tex]

[tex]\frac{14}{u.\cos56}=\frac{u\sin56}{9.8} +\sqrt{(\frac{(u.\sin56)^2}{96.04} -\frac{3.4}{4.9} )}[/tex]

[tex]\frac{196}{u^2.\cos^2 56} +\frac{u^2\sin^2 56}{96.04} -2.857\times \tan56=\frac{u^2\sin^2 56}{96.04} -0.694[/tex]

[tex]u=13.3032\ m.s^{-1}[/tex]

Max height:

[tex]h_m=\frac{(u.\sin 56)^2}{19.6}[/tex]

[tex]h_m=\frac{(13.3032\times \sin56)^2}{19.6}[/tex]

[tex]h_m=6.2059\ m[/tex]

Now the rock hits down the ground 1.6 meters below the level of throw.

Time taken by the rock to fall the gross height [tex]h_g=h_m+h'[/tex]:

[tex]h_g=u_{yt}.t_g+\frac{1}{2} g.t_g^2[/tex]

[tex]7.8059=0+0.5\times 9.8\times t_g^2[/tex]

[tex]t_g=1.2621\ s[/tex]

Time taken to reach the the top of the fence from the top, using eq. (2):

[tex]t_d=\sqrt{(\frac{u_y^2}{96.04} -\frac{3.4}{4.9} )}[/tex]

[tex]t_d=\sqrt{(\frac{(u.\sin56)^2}{96.04} -\frac{3.4}{4.9} )}[/tex]

[tex]t_d=0.7567\ s[/tex]

Time difference between falling from top height and the time taken to reach the top of fence:

[tex]\Delta t=t_g-t_d[/tex]

[tex]\Delta t=1.2621-0.7567[/tex]

[tex]\Delta t=0.5054\ s[/tex]

b)

Now the horizontal distance covered in this time:

[tex]s=u.\cos56\times\Delta t[/tex]

[tex]s=13.3032\times \cos56\times 0.5054[/tex]

[tex]s=3.7597\ m[/tex] is the horizontal distance covered after crossing the fence.

Answer:

[tex]6.67ft/s^2[/tex]

Explanation:

We are given that

Initial velocity=u=18ft/s

Final velocity,v=38ft/s

Time=t=3 s

We have to find the average acceleration over that 3 s period.

We know that

Average acceleration,a=[tex]\frac{v-u}{t}{t}[/tex]

Using the formula

Average acceleration,a=[tex]\frac{38-18}{3}ft/s^2[/tex]

Average acceleration,a=[tex]\frac{20}{3}ft/s^2[/tex]

Average acceleration,a=[tex]6.67ft/s^2[/tex]

Hence, the average acceleration=[tex]6.67ft/s^2[/tex]

Titan is one of Saturn's largest satellites. The molecules on the Saturn's cold moon titan absorb the light from the Sun light because the atmosphere on the titan is at low temperature.Titan is made of thick layers of ice, hence it is relatively cold. If the sunlight reflects from saturns moon Titan, due to the prescence of cold atmosphere, abosrption spectrum arises. So the Spectrum formed by the reflected light from the titan is absorption spectrum

The correct option is C: Absorption spectrum

The visible spectrum of sunlight reflected from Titan, Saturn's moon, would be an absorption spectrum, as Titan's atmosphere absorbs some wavelengths of sunlight. The term 'absorption spectrum' refers to a spectrum produced when light passes through a cool, dilute gas.

The visible spectrum of sunlight reflected from Saturn's moon Titan would be expected to be an absorption spectrum. This is because Titan's atmosphere and surface would absorb some wavelengths of sunlight and reflect the rest, producing an absorption spectrum. There are three types of spectrums: continuous, emission, and absorption. A continuous spectrum is one where all colors (wavelengths) are present without any gaps, which usually represents an ideal black body radiator. An emission spectrum is a spectrum of the electromagnetic radiation emitted by a source. The absorption spectrum, on the other hand, is a spectrum produced when light passes through a cool, dilute gas and atoms in the gas absorb at specific frequencies; since the re-emitted light is unlikely to be emitted in the same direction as the absorbed photon, this gives rise to dark lines (absence of light) in the spectrum.

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Answer:

a. by collisions and mergers of planetesimals.

Explanation:

Inner planets are planets within 1.5 AU distance from the sun. These are called terrestrial planets because they are somewhat similar to Earth, mainly made of rocks.

The main ingredient of these planets are solar nebula and interstellar dust condensation of which leads to formation of small rock particles. These particles come close to each other under in the influence of gravity and other forces. As the mass of the particles increase they form planetesimals, these planetesimals eventually merge to form planets.

Answer:

Acceleration= -0.6466 m/[tex]sec^{2}[/tex]   Downward

Explanation:

Given: Mass M= 945 Kg, Tension T = 8.65 kN = 8650 N and g =9.8 m/[tex]sec^{2}[/tex]

Sol: Weight W = mg = 945 Kg 9.8 m/[tex]sec^{2}[/tex] = 9261 N

this show that T < W so the motion is downward so to find acceleration

mass × acceleration = T - W        (putting values)

945 Kg × a = 8650 N - 9261 N

a= -0.6466 m/[tex]sec^{2}[/tex]  (-ve sign shows the downward direction)

Answer:

- Magnitude of the acceleration = 0.65 m/s²

- The acceleration is directed upwards in the direction of the Tension in the suspending cables.

Explanation:

The force balance on the elevator consists of the Tension in the cable, acting upwards away from the elevator, the weight of the elevator, mg, acting downwards and the 'ma' force responsible for motion.

The direction of this 'ma' force depends on which side of the force balance is lesser or more.

That is, depending on the one that's higher,

ma = T - mg

OR

ma = mg - T

We need to find the higher force.

T = 8.65 KN = 8650 N

mg = 945 × 9.8 = 9261 N

mg > T, meaning the 'ma' force is on the upwards side of the tension, but motion of the elevator is definitely downwards.

Since mg > T,

ma = mg - T = 9261 - 8650 = 611 N

a = 611/m = 611/945 = 0.65 m/s²

Hope this helps!

Answer:

head straight across the river (perpendicular to the bank).

Explanation:

To cross the river in the shortest time first your velocity should be relative to the earth has to have the largest possible component to the bank

suppose,

S be the swimmer

E be the earth

W be the water

[tex]u_{x/y}[/tex]   be the velocity of X relative to Y

resultant velocity relative to E will be:

[tex]u_{S/E}=u_{S/W}+u_{W/E}[/tex]

[tex]u_{W/E}[/tex] is parallel to the bank so,

[tex]u_{S/E}[/tex] has its largest component perpendicular to the bank when [tex]u_{S/W}[/tex] is in that direction

so to cross the river in the shortest time you should straight across the current will then carry you downstream so your path relative to the earth is directed at angle downstream

Final answer:

To cross the river in the shortest time, you must swim perpendicularly to the current. In this case, swimming directly eastward with a speed of 1.5 m/s across a 60 m wide river flowing north at 1.2 m/s will take you across in 40 seconds without being carried downstream.

Explanation:

To cross the river in the shortest time, you must aim to minimize the time spent fighting the water current. The key is to swim in a direction such that your velocity relative to the water combines with the river's velocity to give a resultant path straight across the river. Since the river is flowing north with a speed of 1.2 m/s and your swimming speed relative to the water is 1.5 m/s, the shortest path across would be due east.

If you swim directly eastward, your swimming speed relative to the water ensures that you are moving across the river without being pushed downstream. Thus, the only velocity affecting your eastward crossing is your swimming speed, which is perpendicular to the current. Since the water's current is orthogonal to your motion, it does not affect the time it takes to cross. You'll cross the 60 m wide river in the shortest amount of time by moving at your maximum speed of 1.5 m/s directed perpendicularly to the current.

Considering a swimmer in the given scenario, here's an example to illustrate this concept with numbers:

Therefore, the time taken to cross the river is 40 seconds, and the path taken by the swimmer is perpendicular to the flow of the river, relative to the Earth.

Answer:

The number of electrons that must be moved from one electrode to the other to accomplish this is 1.4 X 10⁹ electrons.

Explanation:

Step 1: calculate the charge on each electrode

Given;

Electric field strength = 2.0 X 10⁶ N/C

The distance between the electrode = 1mm = 1 X 10⁻³ m

Electric field strength (E) = Force (F)/Charge (q)

[tex]E =\frac{Kq}{r^2}[/tex]

where;

E is the electric field strength = 2.0 X 10⁶ N/C

K is coulomb's constant = 8.99 X 10⁹ Nm²/C²

r is the distance between the electrodes = 1 X 10⁻³ m

q is the charge in each electrode = ?

[tex]q = \frac{Er^2}{K} = \frac{(2X10^6)(1X10^{-3})^2}{8.99 X10 ^9}[/tex] = 0.2225 X 10⁻⁹ C

The charge on each electrode is 0.2225 X 10⁻⁹ C

Step 2: calculate the number of electrons to be moved from one electrode to the other.

1 electron contains 1.602 X 10⁻¹⁹ C

So, 0.2225 X 10⁻⁹ C will contain how many electrons ?

= (0.2225 X 10⁻⁹)/(1.602 X 10⁻¹⁹)

= 1.4 X 10⁹ electrons

Therefore, the number of electrons that must be moved from one electrode to the other to accomplish this is 1.4 X 10⁹ electrons.

Final answer:

To create a uniform electric field of 2.0 x 10^6 N/C between two electrodes 1.0 mm apart, a charge of 3.54 x 10^-6 C is required on each electrode. Approximately 2.21 x 10^13 electrons must be moved from one electrode to the other to achieve this.

Explanation:

To solve the immediate question, we first need to calculate the charge required to create a uniform electric field of strength 2.0 x 106 N/C between two rectangular electrodes that are 1.0 mm apart. The relationship between the electric field (E), the charge (Q), the permittivity of free space (ε0), and the distance (d) between the plates is given by the equation E = Q / (ε0 × A), where A is the area of one of the plates. Given the plates have dimensions of 1.0 cm * 2.0 cm, we firstly convert these into meters, giving an area of 0.0002 m2. The permittivity of free space (ε0) is 8.85 x 10-12 C2/N·m2.

Substituting the given values into the formula, we get Q = E × ε0 × A = 2.0 x 106 N/C × 8.85 x 10-12 C2/N·m2 × 0.0002 m2 = 3.54 x 10-6 C. Therefore, this is the charge required on each electrode.

To find the number of electrons that must be moved, we use the relationship between charge and number of electrons, which is given by Q = n × e, where n is the number of electrons and e is the charge of an electron (1.602 x 10-19 Coulombs). So, n = Q / e = 3.54 x 10-6 C / 1.602 x 10-19 C = 2.21 x 1013 electrons. Hence, approximately 2.21 x 1013 electrons need to be moved from one electrode to the other to create the desired electric field.

Answer:

The angle between the electric field and sheet's normal vector is 80.96 degrees.

Explanation:

Given that,

Area of the flat sheet, [tex]A=3.8\ m^2[/tex]

Electric field, E = 10 N/C

Electric flux of the sheet, [tex]\phi=6\ Nm^2/C[/tex]

The electric flux is through the sheet is given by the dot product of electric field and the area vector. It is given by :

[tex]\phi=E{\cdot} A[/tex]

or

[tex]\phi=EA\ cos\theta[/tex]

[tex]\theta[/tex] is the angle between electric field and sheet's normal vector

So,

[tex]cos\theta=\dfrac{\phi}{EA}[/tex]

[tex]cos\theta=\dfrac{6}{10\times 3.8}[/tex]

[tex]\theta=cos^{-1}(0.157)[/tex]

[tex]\theta=80.96^{\circ}[/tex]

So, the angle between the electric field and sheet's normal vector is 80.96 degrees. Hence, this is the required solution.

Final answer:

The angle between the electric field and the flat sheet's normal vector, given the electric flux of 6.0 Nm²/C and field magnitude of 10 N/C, is approximately 81.2 degrees.

Explanation:

The question involves calculating the angle between an electric field and a flat sheet's normal vector, given the electric flux and the field magnitude. The formula for electric flux (Φ) is given by Φ = E * A * cos(θ), where E is the electric field strength, A is the area through which the field lines pass, and θ is the angle between the field and the normal to the surface. In this case, we have the electric flux (Φ = 6.0 Nm²/C), the electric field (E = 10 N/C), and the area (A = 3.8 m²). To find the angle θ, we rearrange the equation to solve for the cosine of the angle: cos(θ) = Φ / (E * A).

Substituting the given values, we get cos(θ) = 6.0 / (10 * 3.8), which simplifies to cos(θ) = 0.1579. Taking the arccosine of both sides, we find θ ≈ arccos(0.1579). By calculating this, we find that θ ≈ 81.2°.

Thus, the angle between the electric field and the sheet's normal vector is approximately 81.2 degrees.

Answer:

The magnitude of the object's average velocity is 9.74 m/s (9.80 m/s without any intermediate rounding).

Explanation:

Hi there!

The average velocity is calculated as the displacement of the object divided by the time it takes the object to do that displacement.

The displacement is calculated as the distance between the final position of the object and the initial position. In this problem, the displacement is equal to the traveled distance because the object travels only in one direction:

a.v = Δx/t

Where:

a.v = average velocity.

Δx = displacement = final position - initial position

t = time

So, let's find the distance traveled while the object was accelerating. For that, we will use this equation:

x = x0 + v0 · t + 1/2 · a · t²

Where:

x = position of the object at time t.

x0 = initial position.

v0 = initial velocity.

t = time.

a = acceleration.

In this case, since the object is initially at rest, v0 = 0. If we place the origin of the frame of reference at the point where the object starts moving, then x0 = 0. So, the equation of the position of the object after a time t will be:

x = 1/2 · a · t²

x = 1/2 · 1.90 m/s² · (6.60 s)²

x = 41.4 m

The object traveled 41.4 m during the first 6.60 s.

Now, let's find the rest of the traveled distance.

When the velocity is constant, a = 0. Then, the equation of position will be:

x = x0 + v · t

Let's place now the origin of the frame of reference at the point where the object starts traveling at constant velocity so that x0 = 0:

x = v · t

The velocity reached by the object during the acceleration phase is calculated as follows:

v = v0 + a · t   (v0 = 0 because the object started from rest)

v = 1.90 m/s² · 6.60 s

v = 12.5 m/s

Then, the distance traveled by the object at a constant velocity will be:

x = 12.5 m/s · 8.50 s

x = 106 m

The total traveled distance in 15.1 s is (106 m + 41.4 m) 147 m.

Then the displacement will be:

Δx = final position - initial position

Δx = 147 m - 0 = 147 m

and the average velocity will be:

a.v = Δx/t

a.v = 147 m / 15.1 s

a.v = 9.74 m/s

The magnitude of the object's average velocity is 9.74 m/s (9.80 m/s without any intermediate rounding).

Answer:

19m/s

22.3 degrees

Explanation:

it is a case aof relative velocity.

the basic for relative velocity vector equation is :

V_b = V_j + V_(b/j)---------------1

V_b: ball velocity relative to ground

V_j : Jaun velocity relative to ground

V_(b/j): ball velocity relative to jaun

reference frame:

We take east and north as +ve x and +ve y

V_(b/j) = V_b - V_j

so for x-axis;

net x-component of V_(b/j) = 12 sin (37) + 0 = 7.22m/s

net y-component of V_(b/j) = 12 cos (37) + 8 = 17.6m/s

magnitude = ((7.22^2)+(17.6^2))^(0.5) = 19 m/s

*direction with respect to Jaun = angle between the vertical (North) and vector V_(b/j)

angle = arctan(7.22/17.6) = 22.3 degrees

The given question is incomplete. The complete question is as follows.

A wave has the mathematical form y = [tex](0.2 m) sin(2\pi ft - \frac{2 \pi x}{\lambda}[/tex]. What is the displacement of a particle at the origin after a time [tex]t = \frac{1}{8T}[/tex]?

Explanation:

Let us assume that at origin, x = 0 and the value of t is given as [tex]\frac{1}{8T}[/tex].

Therefore, we will calculate the value of displacement as follows.

            y = [tex]0.2 \times sin[\frac{2 \pi}{T} \times \frac{1}{8}T - \frac{2 \pi}{\lambda (0)}][/tex]

              = [tex]0.2 sin [\frac{\pi}{4} - 0][/tex]

             = 0.1414 m

             = 0.14 (approx)

Thus, we can conclude that displacement of a particle at the origin after a time [tex]t = \frac{1}{8T}[/tex] is 0.14.

The energy transferred to the car would be different if it were accelerated to the same speed in a shorter time period.

The energy transferred to the car would indeed be different if it were accelerated to the same speed in 5 seconds instead of 10 seconds. This is because the rate of acceleration affects the amount of energy transferred. In the first scenario, the car would experience a lower rate of acceleration over a longer time period, resulting in a smaller energy transfer. In the second scenario, the car would experience a higher rate of acceleration over a shorter time period, resulting in a larger energy transfer.

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Answer:

b. From 33 km/h to 43 km/h

Explanation:

Lets take mass of the car = m

We know that The change kinetic energy KE is give as

[tex]KE=\dfrac{1}{2}m(v^2-u^2)[/tex]

When speed changes from 23 km/h to 33 km/h :

We know that 1 km/h= 0.27 m/s

[tex]KE=\dfrac{1}{2}m(v^2-u^2)[/tex]

[tex]KE=\dfrac{1}{2}\times m((0.27\times 33)^2-(0.27\times 23)^2)[/tex]

KE=  20.412m   J

When speed changes from 33 km/h to 43 km/h :

We know that 1 km/h= 0.27 m/s

[tex]KE=\dfrac{1}{2}m(v^2-u^2)[/tex]

[tex]KE=\dfrac{1}{2}\times m((0.27\times 43)^2-(0.27\times 33)^2)[/tex]

KE=  27.702m   J

Therefore we can say that when speed changes 33 km/h to 43 km/h ,the kinetic energy will changes more.

Answer:

66.375 x 10⁻⁶ C/m

Explanation:

Using Gauss's law which states that the net electric flux (∅) through a closed surface is the ratio of the enclosed charge (Q) to the permittivity (ε₀) of the medium. This can be represented as ;

∅ = Q / ε₀        -----------------(i)

Where;

∅ = 7.5 x 10⁵ Nm²/C

ε₀ = permittivity of free space (which is air, since it is enclosed in a bag) = 8.85 x 10⁻¹² Nm²/C²

Now, let's first get the charge (Q) by substituting the values above into equation (i) as follows;

7.5 x 10⁵ = Q / (8.85 x 10⁻¹²)

Solve for Q;

Q = 7.5 x 10⁵ x 8.85 x 10⁻¹²

Q = 66.375 x 10⁻⁷ C

Now, we can find the linear charge density (L) which is the ratio of the charge(Q) to the length (l) of the rod. i.e

L = Q / l     ----------------------(ii)

Where;

Q = 66.375 x 10⁻⁷ C

l = length of the rod = 10.0cm = 0.1m

Substitute these values into equation (ii) as follows;

L = 66.375 x 10⁻⁷C / 0.1m

L = 66.375 x 10⁻⁶ C/m

Therefore, the linear charge density (charge per unit length) on the rod is 66.375 x 10⁻⁶ C/m.

The linear charge density of the rod is found using Gauss's law and is equal to 6.642 x 10^-5 C/m.

The concept in this question is Gauss's law, which states that the electric flux through any closed surface is equal to the total charge enclosed divided by the permittivity of free space, ε0. Mathematically, it is written as Φ = Q/ε0, where Φ is the electric flux, Q is the charge, and ε0 = 8.854 x 10^-12 C2/N·m2.

In this case, we have Φ = 7.50 x 105 N·m2/C. To find Q, we just rearrange the formula and multiply both sides by ε0. This gives Q = Φ * ε0 = 7.50 x 105 N·m2/C * 8.854 x 10^-12 C2/N·m2 = 6.642 x 10^-6 C.

The linear charge density λ is the total charge Q divided by the total length L of the rod. Here, L = 10.0 cm = 0.1 m, so λ = Q/L = 6.642 x 10^-6 C / 0.1 m = 6.642 x 10^-5 C/m.

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The question is incomplete but an analysis of the situation using  various useful physics concepts can still be made

Answer:

Answer:

[tex]a=24.025\ m/s^2[/tex]    

Explanation:

Given that

Distance from the center ,r= 0.1 m

The angular speed ,ω = 15.5 rad/s

We know that centripetal acceleration is given as

a=ω² r

a=Acceleration

r=Radius

ω=angular speed

a=ω² r

Now by putting the values in the above equation we get

[tex]a=15.5^2\times 0.1\ m/s^2[/tex]

[tex]a=24.025\ m/s^2[/tex]

Therefore the acceleration of the clay will be [tex]a=24.025\ m/s^2[/tex].

Answer:

a. [tex]v = \frac{mu_sm_2g\Delta t}{m_1}[/tex]

b. 21.64 m/s

Explanation:

Let g = 9.81m/s2

a. The weight of the block is product of its mass and gravitational acceleration

[tex]W = m_2g = 7.25*9.81 = 71.1225N[/tex]

which is also the normal force acting on the block from the floor so it stays balanced.

N = 71.225N

The static friction of the block is product of its normal force from the floor and the friction coefficient

[tex]F_s = \mu_sN = \mu_sW = mu_sm_2g[/tex]

For the block to move, the force generated by the impact must be at least equal to the static friction.

[tex]F = F_s = mu_sm_2g[/tex]

The impulse is product of this force and time duration of impact.

[tex]I = F\Delta t = mu_sm_2g\Delta t[/tex]

As impulse is generated by change in momentum of the ball, which is product of its mass and velocity v

[tex]I = \Delta p = m_1\Delta v[/tex]

[tex]mu_sm_2g\Delta t = m_1 v[/tex]

[tex]v = \frac{mu_sm_2g\Delta t}{m_1}[/tex]

b. [tex]v = \frac{mu_sm_2g\Delta t}{m_1} = \frac{0.74*7.25*9.81*0.185}{0.45} = 21.64 m/s[/tex]

From Newton's second law, the minimum velocity the ball must have, to make the block move is 21 m/s

There are two types of collision

In elastic collision, both momentum and energy are conserved.

Given that a baseball of mass m1 = 0.45 kg is thrown at a concrete block m2 = 7.25 kg. The block has a coefficient of static friction of μs = 0.74 between it and the floor. The ball is in contact with the block for t = 0.185 s while it collides elastically.

For the concreate block to move, the force applied must be greater than the friction between the block and the floor.

The frictional force = μN

where N = mg

Friction = 7.25 x 9.8 x 0.72

Friction = 51.156

Let assume that the force applied will be equal to the friction. From Newton's second law,

F = Change in momentum / time taken.

That is

F = [tex]m_{1}[/tex]V / t

since the ball is starting from rest, the initial velocity u = 0

a. The expression for the minimum velocity the ball must have, to make the block move will be

F = [tex]m_{1}[/tex]V / t

Make V the subject of formula

V = Ft / [tex]m_{1}[/tex]

substitute F into the formula

V = μ[tex]m_{2}[/tex]g t / [tex]m_{1}[/tex]

b. The velocity in m/s will be calculated by substituting all the parameters into the formula above.

V = (51.156 x 0.185) / 0.45

V = 9.46386 / 0.45

V = 21 m/s

Therefore, the minimum velocity the ball must have, to make the block move is 21 m/s

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Answer:

The transverse component of acceleration is 26.32 [tex]m/s^2[/tex] where as radial the component of acceleration is 8.77 [tex]m/s^2[/tex]

Explanation:

As per the given data

u=π/4 rad

ω=u'=2 rad/s

α=u''=4 rad/s

[tex]r=e^u[/tex]

So the transverse component of acceleration are given as

[tex]a_{\theta}=(ru''+2r'u')\\[/tex]

Here

[tex]r=e^u\\r=e^{\pi/4}\\r=2.1932 m[/tex]

[tex]r'=e^u.u'\\r'=2.1932 \times 2\\r'=4.3864 m[/tex]

So

[tex]a_{\theta}=(ru''+2r'u')\\a_{\theta}=(2.1932\times 4+2\times 4.3864 \times 2)\\a_{\theta}=26.32 m/s\\[/tex]

The transverse component of acceleration is 26.32 [tex]m/s^2[/tex]

The radial component is given as

[tex]a_r=r''-r\theta'^2[/tex]

Here

[tex]r''=e^u.u'^2+e^u u''\\r''=2.1932 \times (2)^2+2.1932\times 4\\r''=17.5456 m[/tex]

So

[tex]a_r=r''-ru'^2\\a_r=17.5456-2.1932\times (2)^2\\a_r=8.7728 m/s^2[/tex]

The radial component of acceleration is 8.77 [tex]m/s^2[/tex]

The radial and transverse components of the peg's acceleration at the given instant are both 4eu m/s^2.

In circular motion, there are two components of acceleration: the radial component (also called centripetal acceleration), directed towards the center of the circle, and the transverse component (also called tangential or azimuthal acceleration), which is perpendicular to the radial component and in the direction of increasing angle.

Given that r = eu, and u = p4 rad, u# = 2 rad/s (angular velocity), u$ = 4 rad/s^2 (angular acceleration), we can calculate these components as follows:

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Answer:

The number of vacancies per cubic meter is 1.18 X 10²⁴ m⁻³

Explanation:

[tex]N_v = N*e[^{-\frac{Q_v}{KT}}] = \frac{N_A*\rho _F_e}{A_F_e}e[^-\frac{Q_v}{KT}}][/tex]

where;

N[tex]_A[/tex] is the number of atoms in iron = 6.022 X 10²³ atoms/mol

ρFe is the density of iron = 7.65 g/cm3

AFe is the atomic weight of iron = 55.85 g/mol

Qv is the energy vacancy formation = 1.08 eV/atom

K is Boltzmann constant = 8.62 X 10⁻⁶ k⁻¹

T is the temperature = 850 °C = 1123 k

Substituting these values in the above equation, gives

[tex]N_v = \frac{6.022 X 10^{23}*7.65}{55.85}e[^-\frac{1.08}{8.62 X10^{-5}*1123}}]\\\\N_v = 8.2486X10^{22}*e^{(-11.1567)}\\\\N_v = 8.2486X10^{22}*1.4279 X 10^{-5}\\\\N_v = 1.18 X 10^{18}cm^{-3} = 1.18 X 10^{24}m^{-3}[/tex]

Therefore, the number of vacancies per cubic meter is 1.18 X 10²⁴ m⁻³

The number of vacancies will be "1.18 × 10²⁴ m⁻³".

According to the question,

Number of atoms, [tex]N_A[/tex] = 6.022 × 10²³ atoms/mol

Iron's density, ρFe = 7.65 g/cm³

Iron's atomic weight, AFe = 55.85 g/mol

Energy vacancy formation, Qv = 1.08 eV/atom

Boltzmann constant, K = 8.62 × 10⁻⁶ k⁻¹

Temperature, T = 850°C or, 1123 K

We know the formula,

→ [tex]N_v[/tex] = N × e [[tex]-\frac{Qv}{KT}[/tex]]

        = [tex]\frac{N_A\times \rho Fe}{AFe}[/tex] e [[tex]-\frac{Qv}{KT}[/tex]]

By substituting the above values, we get

        = [tex]\frac{6.022\times 10^{23}\times 7.65}{55.85}[/tex] e [[tex]- \frac{1.08}{8.62\times 10^{-5}\times 1123}[/tex]]

        = 8.2486 × 10²² × [tex]e^{(-11.1567)}[/tex]

        = 8.2486 × 10²² × 1.4279 × 10⁻⁵

        = 1.18 × 10¹⁸ cm⁻³ or,

        = 1.18 × 10²⁴ m⁻³

Thus the answer above is correct.

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Answer:

The horizontal force is 780 N.

Explanation:

Given that,

Mass of sprinter = 52 kg

Suppose A world-class sprinters can accelerate out of the starting blocks with an acceleration that is nearly horizontal and has magnitude 15 m/s².

We need to calculate the horizontal force

Using formula of force

[tex]F = ma[/tex]

Where, m = mass of sprinter

a = acceleration

Put the value into the formula

[tex]F=52\times15[/tex]

[tex]F=780\ N[/tex]

Hence, The horizontal force is 780 N.

Answer:

E=[tex]\frac{KQ}{2\sqrt 2a^2}[/tex]

Explanation:

We are given that

Charge on ring= Q

Radius of ring=a

We have to find the magnitude of electric filed on the axis at distance a from the ring's center.

We know that the electric field at distance x from the center of ring of radius R is given by

[tex]E=\frac{kQx}{(R^2+x^2)^{\frac{3}{2}}}[/tex]

Substitute x=a and R=a

Then, we get

[tex]E=\frac{KQa}{(a^2+a^2)^{\frac{3}{2}}}[/tex]

[tex]E=\frac{KQa}{(2a^2)^{\frac{3}{2}}}[/tex]

[tex]E=\frac{KQa}{2\sqrt 2a^3}[/tex]

[tex]E=\frac{KQ}{2\sqrt 2a^2}[/tex]

Where K=[tex]9\times 10^9 Nm^2/C^2[/tex]

Hence, the magnitude of the electric filed due to charged ring on the axis of ring at distance a from the ring's center=[tex]\frac{KQ}{2\sqrt 2a^2}[/tex]

The magnitude of the electric field due to a charged ring of radius "a" and total charge "Q", at a point on the ring axis a distance "a" from the ring's center is E = Q/[8√2πε₀a²]

The electric field due to a charged ring E is given by

E = Qz/4πε₀[√(z² + R²)]³ where

Given that for this ring R = a and z = a, substituting these values into E, the magnitude of the electric field at a is given by

E = Qz/4πε₀[√(z² + R²)]³

E = Qa/4πε₀[√(a² + a²)]³

E = Qa/4πε₀[√(2a²)]³

E = Qa/4πε₀[2√2a³]

E = Q/[8πε₀√2a²]

E = Q/[8√2πε₀a²]

So, the magnitude of the electric field due to a charged ring of radius "a" and total charge "Q", at a point on the ring axis a distance "a" from the ring's center is E = Q/[8√2πε₀a²]

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Answer: The value of spring constant is 121.9 N/m

Explanation:

Force is defined as the mass multiplied by the acceleration of the object.

[tex]F=m\times g[/tex]

where,

F = force exerted on the object  = ?

m = mass of the object  = 0.46 kg

g = acceleration due to gravity = [tex]9.81m/s^2[/tex]

Putting values in above equation, we get:

[tex]F=0.46kg\times 9.81m/s^2=4.51N[/tex]

To calculate the spring constant, we use the equation:

[tex]F=k\times x[/tex]

where,

F = force exerted on the spring = 4.51 N

k = spring constant = ?

x = length of the spring = 3.7 cm = 0.037 m     (Conversion factor:  1 m = 100 cm)

Putting values in above equation, we get:

[tex]4.51N=k\times 0.037m\\\\k=\frac{4.51N}{0.037m}=121.9N/m[/tex]

Hence, the value of spring constant is 121.9 N/m

Answer:

[tex]heigth=83.44m[/tex]

Explanation:

Given data

Baseball mass m₁=0.15 kg

initial speed v₁=0

Bullet mass m₂=0.032 kg

final speed v₂=230 m/s

To find

height h=?

Solution

From conservation of momentum we know that

[tex]m_{1}v_{1}+m_{2}v_{2}=(m_{1}+m_{2})v\\ (0.032kg)(230m/s)+(0.15kg)(0m/s)=(0.15kg+0.032kg)v\\7.36+0=0.182v\\v=7.36/0.182\\v=40.44m/s[/tex]

Now from the conservation of mechanical energy

[tex]P.E=K.E\\mgh=(1/2)mv^{2}\\ gh=(1/2)v^{2}\\ (9.8m/s^{2} )h=(1/2)(40.44m/s)^{2}\\ (9.8m/s^{2} )h=(817.7m^{2} /s^{2} )\\h=(817.7m^{2} /s^{2} )/9.8m/s^{2}\\ heigth=83.44m[/tex]

Final answer:

To determine the rise after a bullet embeds itself in a baseball, use conservation of momentum to calculate the final velocity of the combined mass, then use conservation of energy to find the maximum height the system achieves.

Explanation:

The student is asking about the vertical rise of a combined system of a bullet and a baseball after a collision in which the bullet embeds itself into the baseball. To determine the height to which the combined mass rises, we first need to apply the principle of conservation of momentum to find the velocity of the combined mass right after the collision. Then we use the conservation of energy to find the maximum height the combined mass reaches.

The conservation of momentum before and after the collision gives us:

Initial momentum = mass of bullet × velocity of the bullet

Final momentum = (mass of bullet + mass of baseball) × final velocity

Since initial and final momentum are equal (assuming no external net forces), we can set them equal to each other to solve for the final velocity.

Next, to find the maximum height, we use the conservation of energy, where the initial kinetic energy of the combined mass is converted to gravitational potential energy at the peak of its rise. This gives us:

Kinetic energy = (1/2) × (mass of bullet + mass of baseball) × (final velocity)2

Potential energy at height = (mass of bullet + mass of baseball) × g (acceleration due to gravity) × height

By equating the kinetic energy right after the collision to the potential energy at the maximum height, we can solve for the height.

Explanation:

p1:                           Ww  x   ww

gametes:                       w,W      w,w

                                    .......cross......

F1:                             Ww (2) and ww (2)

                               (white)         (black)

therefore,

genotype  ratio -    1: 1

phenotype ratio-    1: 1

The offspring of a cross between a WW (white) and ww (black) parent will all have the genotype Ww and a white phenotype, resulting in a 100% phenotype ratio for white and a 100% genotype ratio for Ww.

When analyzing the cross between a parent with the genotype WW (white) and another with the genotype ww (black), we consider the basic principles of Mendelian genetics. Since white WW is homozygous dominant and black ww is homozygous recessive, all the offspring will have the genotype Ww, making them all white, due to the complete dominance of the white allele.

The phenotype ratio for this cross would be 100% white offspring. The genotype ratio would be 100% heterozygous Ww, because each offspring inherits one W allele from the white parent and one w allele from the black parent.

The force exerted by each rope is approximately 230 N.

Let's denote the magnitude of the force exerted by each rope as F. Given that the resultant pull is 372 N directly upward, we can use vector decomposition to find the individual forces exerted by each rope.

The forces F acting at an angle of 72° between them can be decomposed into their horizontal and vertical components as follows:

[tex]\[ F_{\text{horizontal}} = F \cdot \cos(72^\circ) \]\\\ F_{\text{vertical}} = F \cdot \sin(72^\circ) \][/tex]

The vertical components of the forces add up to the resultant pull, so:

[tex]\[ 2 \cdot F_{\text{vertical}} = 372 \, \text{N} \]\\\\\ F_{\text{vertical}} = \frac{372 \, \text{N}}{2} \\\\= 186 \, \text{N} \][/tex]

Now, we can use the vertical component to find F:

[tex]\rm \[ F = \frac{F_{\text{vertical}}}{\sin(72^\circ)} \][/tex]

Substitute the known values:

[tex]\rm \[ F = \frac{186 \, \text{N}}{\sin(72^\circ)} \][/tex]

Calculate F:

[tex]\rm \[ F \approx 230 \, \text{N} \][/tex]

So, the force exerted by each rope is approximately 230 N.

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To find the pull exerted by each rope with an angle of 72°, we can use vector components. Each rope exerts a pull of approximately 436.2 N at an angle of 72° for both the vertical and horizontal components.

To find the pull exerted by each rope, we can use the concept of vector components. Let's call the magnitude of each pull T. Since the resultant pull is 372 N directly upward, we can calculate the vertical component of each rope's pull using the equation T*sin(72°) = 372 N.

Solving for T, we find that the magnitude of each rope's pull is approximately 436.2 N. To find the horizontal component of each rope's pull, we use the equation T*cos(72°).

Since the forces are equal-magnitude and have the same angle between them, each rope exerts the same pull of approximately 436.2 N at an angle of 72° for both the vertical and horizontal components of the pull.

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Maximum shear stress in a round steel bar with a groove, subjected to reversed torque, is found by computing the nominal shear stress and multiplying it by the stress concentration factor of the groove. The stress concentration factor must be known to complete the calculation.

To determine the maximum shear stress in an AISI 1040 CD steel round bar which is subjected to purely reversed torque of 1800 lbf∙in, and has a groove machined into it, the effect of the shape modification by the groove needs to be considered. This groove effect is described using stress concentration factor (Kt), which shows the increase in maximum stress over the nominal stress because of the change in geometry.

The method involves determining the nominal shear stress (τnom) which equals the torque (T) divided by the polar moment of inertia (J) given as J = (π * (d/2)^4)/2 for a round rod. Then, multiply τnom by the stress concentration factor of the groove to find the maximum shear stress τmax = Kt * τnom.

However, I see the Kt value for the particular groove shape and size is not provided. This value is usually looked up in standard tables or calculated using specific formulas/fixtures based on the groove's size and shape. Once Kt is known, you can compute the maximum shear stress precisely.

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