A 0.23 kg mass on a spring vibrates with amplitude 25 cm and frequency 1.7 Hz. Calculate (b) the speed at which the mass passes through equilibrium and (b) the total energy of the oscillation. (Answers: 0.82 J, 2.7 m/s).

Answer:

By dropping a pencil from a certain fixed height again and again it will make the target super messay with marks of dot everywhere on the target and some even out side the target.

Explanation:

Answer:

A molecule is a combination of two or more atoms. The correct option is B.

Explanation:

Molecules are the smallest particle of a chemical compound that are made up of two or more atoms which are held together by a chemical bond. The chemical bonds are usually formed due to sharing and transfer of electrons among the atoms. Examples of molecules in chemistry includes:

- water molecule ( H2O)

- table salt ( NaCl)

- CaCl

Electric field due to uniformly charged metal plate is given by,

[tex]E = \frac{\sigma}{(2\epsilon_0)}[/tex]

Here,

[tex]\sigma[/tex] = Charge density

[tex]\epsilon_0 =[/tex]  Vacuum Permittivity

Our values are,

[tex]\sigma = 0.75 muC/m^2 = 0.75*10^-6 C/m^2[/tex]

[tex]\epsilon_0 = 8.85*10^-12 F\cdot m^{-1}[/tex]

Replacing we have,

[tex]E = \frac{(0.75*10^-6)}{(2*8.85*10^-12)}[/tex]

[tex]F = 42372.88N/C[/tex]

Now we have the relation where energy is equal to the change of the potential in a certain distance, then

[tex]E = \frac{V}{d}[/tex]

Rearranging for the distance

[tex]d = \frac{V}{E}[/tex]

[tex]d = \frac{100}{42372.88}[/tex]

[tex]d = 0.00236m[/tex]

[tex]d = 2.36mm[/tex]

Therefore the distance is 2.36mm

Answer:

The time it takes the proton to return to the horizontal plane is 7.83 X10⁻⁷ s

Explanation:

From Newton's second law, F = mg and also from coulomb's law F= Eq

Dividing both equations by mass;

F/m = Eq/m = mg/m, then

g = Eq/m --------equation 1

Again, in a projectile motion, the time of flight (T) is given as

T = (2usinθ/g) ---------equation 2

Substitute in the value of g into equation 2

[tex]T = \frac{2usin \theta}{\frac{Eq}{m}} =\frac{m* 2usin \theta}{Eq}[/tex]

Charge of proton = 1.6 X 10⁻¹⁹ C

Mass of proton = 1.67 X 10⁻²⁷ kg

E is given as 400 N/C, u = 3.0 × 10⁴ m/s and θ = 30°

Solving for T;

[tex]T = \frac{(1.67X10^{-27}* 2*3X10^4sin 30}{400*1.6X10^{-19}}[/tex]

T = 7.83 X10⁻⁷ s

Final answer:

The force exerted on a proton in a magnetic field can be calculated using the given formula with specified values.

Explanation:

To find the time it takes for the proton to return to the horizontal plane, we need to consider the motion in both the horizontal and vertical directions separately.

Given:

[tex]- Initial velocity (\(v_0\)) of the proton = \(3.0 \times 10^4\) m/s\\- Launch angle (\(\theta\)) = 30°\\- Electric field (\(E\)) = 400 N/C\\- We'll assume the gravitational acceleration (\(g\)) as \(9.8 \, \text{m/s}^2\) downward.[/tex]

Vertical Motion:

In the vertical direction, the proton undergoes uniformly accelerated motion under gravity.

Using the formula for vertical motion:

[tex]\[ v = u + at \] \\where: \\- \( v \) is the final velocity (which is 0 when the proton returns to the horizontal plane),\\- \( u \) is the initial vertical velocity,\\- \( a \) is the acceleration due to gravity,\\- \( t \) is the time taken.\\[/tex]

We can resolve the initial velocity into vertical and horizontal components:

[tex]\[ u_y = v_0 \sin(\theta) \]\[ u_y = 3.0 \times 10^4 \times \sin(30^\circ) \]\[ u_y \approx 3.0 \times 10^4 \times 0.5 \]\[ u_y = 1.5 \times 10^4 \, \text{m/s} \][/tex]

Now, let's find the time it takes for the vertical velocity to become zero:

[tex]\[ 0 = u_y - gt \]\[ t = \frac{u_y}{g} \]\[ t = \frac{1.5 \times 10^4}{9.8} \]\[ t \approx 1530.61 \, \text{s} \][/tex]

Horizontal Motion:

In the horizontal direction, there is no acceleration or deceleration acting on the proton. So, the time taken for horizontal motion is the same as the time taken for vertical motion.

The time it takes for the proton to return to the horizontal plane is approximately [tex]\( 1530.61 \) seconds.[/tex]

Answer:

67 dB

Explanation:

given,

Sound in afternoon = 80 dB

Intensity of car,I₀ = 100 cars/ minute

Sound in the night = ?

Intensity of car,I = 5 car/minutes

using formula for sound calculation

 [tex]\Delta \beta = 10 log(\dfrac{I}{I_0})[/tex]

 [tex]\Delta \beta = 10 log(\dfrac{5}{100})[/tex]

 [tex]\Delta \beta = 10 log(\dfrac{1}{20})[/tex]

 [tex]\Delta \beta = 10 log(0.05)[/tex]

 [tex]\Delta \beta = 10\times -1.30[/tex]

 [tex]\Delta \beta = -13\ dB[/tex]

The late night decibel is equal to 80 dB - 13 dB = 67 dB

The radius of a neutron star with a mass 10 times that of the Sun is approximately 29.62 km.

To calculate the radius of a neutron star with a mass 10 times that of the Sun, we can use the equation for the Schwarzschild radius:

R = 2GM / c^2

Where R is the radius, G is the gravitational constant, M is the mass, and c is the speed of light. Plugging in the values, we get:

R = 2 * 6.67x10^-11 (m^3/kg/s^2) * (10 * 1.99x10^30 kg) / (3x10^8 m/s)^2

R ≈ 29.62 km

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Answer:

0.24825 N

0.0000238701923077 kgm²

Explanation:

m = Mass of yo yo = 0.075 kg

a = Acceleration = 6.5 m/s²

g = Acceleration due to gravity = 9.81 m/s²

The net force is given by

[tex]F_n=mg-T[/tex]

[tex]\Rightarrow T=mg-ma[/tex]

[tex]\Rightarrow T=m(g-a)[/tex]

[tex]\Rightarrow T=0.075(9.81-6.5)[/tex]

[tex]\Rightarrow T=0.24825\ N[/tex]

The tension in the string is 0.24825 N

Angular acceleration is given by

[tex]\alpha=\dfrac{a}{r}\\\Rightarrow \alpha=\dfrac{6.5}{2.5\times 10^{-2}}\\\Rightarrow \alpha=260\ rad/s^2[/tex]

Torque is given by

[tex]\tau=I\alpha\\\Rightarrow Tr=I\alpha\\\Rightarrow I=\dfrac{Tr}{\alpha}\\\Rightarrow I=\dfrac{0.24825\times 2.5\times 10^{-2}}{260}\\\Rightarrow I=0.0000238701923077\ kgm^2[/tex]

The moment of inertia is 0.0000238701923077 kgm²

The tension in the string is equal to 0.2475 Newton.

Given the following data:

To determine the tension in the string:

First of all, we would determine the downward force applied by the yo-yo's weight:

[tex]F_y = mg[/tex]

Where:

Substituting the given parameters into the formula, we have;

[tex]F_y = 0.075 \times 9.8\\\\F_y = 0.735 \; Newton[/tex]

Next, we would determine the force acting on the string:

[tex]F_s = 0.075 \times 6.5\\\\F_s = 0.4875\;Newton[/tex]

Now, we can find the tension in the spring:

[tex]Tension = F_y - F_s\\\\Tension = 0.735 - 0.4875[/tex]

Tension = 0.2475 Newton.

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Answer:

The person is 187[m] farther and 70° south to east.

Explanation:

We can solve this problem by drawing a sketch of the location of the person and the truck, then we will draw the displacement vectors and finally the length of the vector and the direction of the vector will be measured in order to give the correct indication of where the person will have to move.

First we establish an origin of a coordinate system.

We can see in the attached schema that the red vector is the displacement vector from the last point to where the truck is located.

The length of the vector is 187 [m], and the direction is 70 degrees south to East.

This problem involves using vectors and trigonometry to calculate the direct distance and direction to the truck from the new location after walking due west. This is achieved by adding the horizontal and vertical components of the vectors representing the initial location and the walking path.

To solve this, one may use vectors and trigonometry. Initially, you are 122.0m from your truck, 58.0 degrees east of south. This can be treated as a vector from your truck to your original location. Then, you walk 72.0m due west, which is another vector in the opposite direction.

To find the resulting vector, i.e., the direct distance and direction to the truck from your new location, we have to add these vectors. While the mathematics is somewhat complex, the concept involves adding the horizontal (east-west) and vertical (north-south) components of each vector. Once the resulting vector is calculated, the remaining distance to the truck can be found from its magnitude, and the direction from its angle relative to south.

This involves math calculations, including trigonometry and Pythagorean theorem.

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The statement is true due to the Doppler Effect. As you move away from the radio transmitter, the radio signal you receive appears to have been shifted to longer wavelengths due to the changes in the observer-source distance.

The statement, 'As you drive away from a radio transmitter, the radio signal you receive from the station is shifted to longer wavelengths,' is True. This is due to a phenomenon known as the Doppler Effect.

The Doppler Effect explains that the observed wavelength of electromagnetic radiation is longer (red shift) when the source moves away from the observer. This means the wavelength of the radiation from the radio station would appear to increase (shift to a longer wavelength) as you drive away from it.

To maintain that same energy level required for transmission, the radio station emits waves at a higher frequency (shorter wavelengths). But, as you move away, the radio waves appear to have a lower frequency (longer wavelengths). This happens because the waves get 'stretched out' as the distance between you (the observer) and the radio station (the source) increases.

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Answer:

A) [tex]k=34867.3384\ N.m^{-1}[/tex]

B) [tex]\omega'\approx84\ Hz[/tex]

Explanation:

Given:

mass of car, [tex]m=1380\ kg[/tex]

A)

frequency of spring oscillation, [tex]f=1.6\ Hz[/tex]

We knkow the formula for spring oscillation frequency:

[tex]\omega=2\pi.f[/tex]

[tex]\Rightarrow \sqrt{\frac{k_{eq}}{m} } =2\pi.f[/tex]

[tex]\sqrt{\frac{k_{eq}}{1380} } =2\times \pi\times 1.6[/tex]

[tex]k_{eq}=139469.3537\ N.m^{-1}[/tex]

Now as we know that the springs are in parallel and their stiffness constant gets added up in parallel.

So, the stiffness of each spring is (as they are identical):

[tex]k=\frac{k_{eq}}{4}[/tex]

[tex]k=\frac{139469.3537}{4}[/tex]

[tex]k=34867.3384\ N.m^{-1}[/tex]

B)

given that 4 passengers of mass 70 kg each are in the car, then the oscillation frequency:

[tex]\omega'=\sqrt{\frac{k_{eq}}{(m+70\times 4)} }[/tex]

[tex]\omega'=\sqrt{\frac{139469.3537}{(1380+280)} }[/tex]

[tex]\omega'\approx84\ Hz[/tex]

Answer:

b. Throwing some cargo out of the boat

Explanation:

Using the Newton's third law of motion which states that every action has an equal and opposite reaction.

So when we are on the boat and we throw some mass in a direction away from the boat out of it then we are imparting the force to the floating boat by the law of conservation of momentum as well.

The relation can be mathematically expressed as:

[tex]m_c.v_c=m_b.v_b[/tex]

where:

[tex]m_c=[/tex] mass of the cargo

[tex]v_c=[/tex] velocity of throwing the cargo

[tex]m_b=[/tex] mass of the whole boat including all that floats on it

[tex]v_b=[/tex] velocity of the boat system

Throwing some cargo out of the boat will make the boat start moving.

This law states that action and reaction are equal and opposite.

The total momentum of an isolated system is always conserved.

Throwing some cargo out of the boat will change the velocity of the cargo initially at rest and to conserve the linear momentum, the velocity of the boat will change as well.

Thus, throwing some cargo out of the boat will make the boat start moving.

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Answer:

Part A:

[tex]d_{min} = \frac{v_0^2}{2g\mu_s}[/tex]

Part B:

[tex]d_{min} = 6.37~m[/tex]

Explanation:

Part A:

We should determine the free-body diagram of the small box.

For the first box, the only force exerted to the box is the static friction force in the direction of the motion.

(The direction of the static friction is always confusing to the students. The wrong idea is that the static friction is in the opposite direction with the motion. However, if you look at the Newton's Second Law, it states that the net force acting on an object is equal to the mass times acceleration. And in this case acceleration of the total system is equal to that of the small box, since it sits on the larger box.)

We can use the equations of kinematics to find the minimum distance to stop without the small box slipping.

[tex]v^2 = v_0^2 + 2a(\Delta x)\\0 = v_0^2 + 2ad_{min}[/tex]

The acceleration can be found by Newton's Second Law:

[tex]F = ma\\mg\mu_s = m(-a)\\a = -g\mu_s[/tex]

The negative sign comes from the fact that in order for the boxes to stop they have to apply a negative acceleration.

Now, we can combine the two equations to find the distance x:

[tex]0 = v_0^2 + 2ad_{min} = v_0^2 + 2(-g\mu_s)d_{min}\\d_{min} = \frac{v_0^2}{2g\mu_s}[/tex]

Part B:

We can apply the above formula to the truck and file cabinet.

[tex]d_{min} = \frac{v_0^2}{2g\mu_s} = \frac{10^2}{2(9.8)(0.80)} = 6.37~m[/tex]

The expression for the shortest distance which the large box can stop without the small box slipping is [tex]d_{min} = \frac{v_0^2 }{2\mu_s g}[/tex]

The shortest distance in which the pickup can stop without the file cabinet sliding is 6.38 m.

The given parameters;

The expression for the shortest distance which the large box can stop without the small box slipping is calculated as follows;

Apply work-energy theorem;

[tex]\mu_s F \times d_{min} = \frac{1}{2} Mv_0^2\\\\\mu_s (Mg)d_{min} = \frac{1}{2} Mv_0^2\\\\\mu_s gd_{min} = \frac{v_0^2}{2} \\\\d_{min} = \frac{v_0^2}{2\mu_s g}\\\\[/tex]

At the given speed and coefficient of static friction, the shortest distance in which the pickup can stop without the file cabinet sliding is calculated as;

[tex]d_{min} = \frac{v^2}{2\mu_s g} = \frac{10^2 }{2\times 0.8 \times 9.8} = 6.38 \ m[/tex]

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Answer:

g/9

Explanation:

length of the pendulum = L

time period on the earth = T

Time period on the planet = 3T

Let the acceleration due to gravity on the earth is g and on the planet is g'.

Use the formula for the time period of a simple pendulum for the time period on earth

[tex]T=2\pi \sqrt{\frac{L}{g}}[/tex]     .... (1)

Time period on the surface of planet is

[tex]3T=2\pi \sqrt{\frac{L}{g'}}[/tex]      .... (2)

Divide equation (2) by equation (1)

[tex]\frac{3T}{T}= \sqrt{\frac{g}{g'}}[/tex]

g' = g/9

Thus, the acceleration due to gravity on the planet is g /9

Options

1. the same as in the first.

2.greater than in the first

3.less than in the first.

4.Unable to determine

Answer

The answer is 2. Greater than in the first

Explanation

The speed of a wave v is related to its wavelength λ by the formulav=f λ, where f is the frequency of the wave. The frequency will not change when the wave passes into a second medium, so

λ2>λ1

Fλ1>fλ2

Since f>0

And V2>v1

Answer:

A. 8.175 N

B. 49.05 N

Explanation:

A.

Acceleration due to gravity (moon) = 1/6 * (acceleration due to gravity (earth)

Acceleration due to gravity (earth) = 9.81 m/s2

Acceleration due to gravity (moon) = 9.81/6

= 1.635 m/s

Weight. Fm = acceleration due to gravity * mass

= 1.635 * 5

= 8.175 N

B. Acceleration due to gravity (earth) * mass = Fe

= 9.81 * 5

= 49.05 N

(i) 625 V/m

(ii) 2876.25 x 10⁻¹² C

(iii) 0.000575.25 x 10⁻⁶ F

(i) The electric field (E) between the plates of a parallel plate capacitor is related to the potential difference (V) between the plates and the distance (d) of separation between the plates as follows;

E = V / d        ----------------(i)

From the question;

V = 5.0V

d = 0.80cm =  0.008m

Substitute these values into equation (i) as follows;

E = 5.0 / 0.008

Solve for E;

E = 625 V/m

Therefore, the electric field in the region between the two plates is 625 V/m.

(ii) To make things easier, let's calculate the capacitance of the parallel plates first.

The capacitance (C) of a parallel plate capacitor is given as;

C = A x ε₀ / d         --------------------------(ii)

Where;

A = Area of either of the plates of the capacitor = 0.52m²

ε₀ = permittivity of free space = 8.85 x 10⁻¹² F/m

d = distance between the plates =  0.8cm = 0.008m

Substitute these values into equation (ii) as follows;

C = 0.52 x 8.85 x 10⁻¹² / 0.008

Solve for C;

C = 575.25 x 10⁻¹² F

The capacitance (C) is related to potential difference (V) and charge (Q) on the plates as follows;

Q = C x V            -------------------------(iii)

Where;

C = 575.25 x 10⁻¹² F

V = 5.0V

Substitute these values into equation (iii)

Q = 575.25 x 10⁻¹² x 5

Q = 2876.25 x 10⁻¹² C

Therefore, the charge on the plates is 2876.25 x 10⁻¹² C

(iii) The capacitance (C) of the parallel plates has been calculated in (ii) above.

Its value is 575.25 x 10⁻¹² F = 0.000575.25 x 10⁻⁶F

The electric field in the region between two plates is 625 V/m. The capacitance of the plates is 0.0572 nanoFarads and the accumulated charge Q on each plate is 2.86 x 10^-10 Coulombs.

The electric field between two plates can be calculated using the formula E = V/d, where V represents voltage and d represents the distance between plates. So, the electric field E would be 5.0 V / 0.008 m = 625 N/C or 625 V/m.  The charge Q on the plates can be determined using the formula Q = CV, where C is the capacitance and V is the voltage. To find the charge, we first need to calculate the capacitance. The capacitance of the parallel plates can be calculated using the formula C = ε0*(A/d), where ε0 is the permittivity of free space (8.85 x 10^-12 F/m), A is the area and d is the distance between the plates. The capacitance then would be 8.85 x 10^-12 F/m * 0.52 m^2 / 0.008 m = 5.72 x 10^-11 F or 0.0572 nF (nanoFarads). So, after finding the capacitance, we can now calculate the charge Q, which is Q = CV = 5.72 x 10^-11 F * 5.0 V = 2.86 x 10^-10 C (Coulombs).

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Answer:

The change in the electric potential energy is [tex]-3.2\times10^{-16}\ J[/tex]

Explanation:

Given that,

Distance [tex]d=4.30\times10^{-11}\ m[/tex]

suppose, Two particles with charges +6 e and -10 e are initially very far apart

We need to calculate the change in the electric potential energy

Using formula of energy

[tex]\text{electric potential energy}=\text{final electric potential energy-initial electric potential energy}[/tex]

[tex]EPE=EPE_{f}-EPE_{i}[/tex]

Here, initial electric potential energy= 0

final electric potential energy [tex]EPE_{f}=\dfrac{kq_{1}q_{2}}{r_{2}^2}[/tex]

Put the value into the formula

[tex]EPE=\dfrac{kq_{1}q_{2}}{r_{2}^2}+0[/tex]

Put the value into the formula

[tex]EPE=\dfrac{9\times10^{9}\times6\times1.6\times10^{-19}\times(-10\times1.6\times10^{-19})}{(4.30\times10^{-11})}[/tex]

[tex]EPE=-3.2\times10^{-16}\ J[/tex]

Hence, The change in the electric potential energy is [tex]-3.2\times10^{-16}\ J[/tex]

This is an incomplete question, here is a complete question.

Two particles with charges +6 e⁻ and -10 e⁻ are initially very far apart (effectively an infinite distance apart). They are then fixed at positions that are 4.30 × 10⁻¹¹ m apart. What is EPE(final) - EPE(initial), which is the change in the electric potential energy?

Answer : The change in the electric potential energy is, [tex]-2.92\times 10^{-6}J[/tex]

Explanation : Given,

Formula used for electric potential energy of the two charges when they are separated is:

[tex]EPE=\frac{1}{4\pi \epsilon_0}\times {\frac{q_1\times q_2}{r^2}[/tex]

[tex]EPE=\frac{k\times q_1\times q_2}{r^2}[/tex]

where,

EPE = electric potential energy

k = [tex]\frac{1}{4\pi \epsilon_0}=8.99\times 10^9[/tex]

[tex]q_1[/tex] = charge on 1st particle = +6 e⁻ = [tex]6\times 10^{-19}C[/tex]

[tex]q_2[/tex] = charge on 2nd particle = -10 e⁻ = [tex]-10\times 10^{-19}C[/tex]

r = distance between two charges = [tex]4.30\times 10^{-11}m[/tex]

Now put all the given values in the above formula, we get:

[tex]EPE=\frac{(8.99\times 10^9)\times (6\times 10^{-19})\times (-10\times 10^{-19})}{(4.30\times 10^{-11})^2}[/tex]

[tex]EPE=-2.92\times 10^{-6}J[/tex]

Initially EPE = 0 J

Thus, [tex]EPE_{final}-EPE_{initial}=-2.92\times 10^{-6}J[/tex]

The positive sign indicate the attractive force and negative sign indicate the repulsive force.

Thus, the change in the electric potential energy is, [tex]-2.92\times 10^{-6}J[/tex]

To calculate the tension in the cable required to stop the elevator, we multiply the mass of the elevator by its acceleration. The tension in the cable is -4356 N, indicating it acts in the opposite direction of the weight of the elevator.

To calculate the tension in the cable required to stop the elevator over a distance of 3.4 m, we need to consider the force required to decelerate the elevator from its maximum speed of 3.3 m/s to a stop. The tension in the cable must equal the force needed to stop the elevator, which is equal to the mass of the elevator multiplied by its acceleration. The mass of the elevator, including occupants, is given as 1320 kg. Since the elevator is going down, its acceleration will be negative. Therefore, the tension in the cable can be calculated using the formula:

Tension = mass * acceleration = 1320 kg * (-3.3 m/s^2) = -4356 N

Therefore, the tension in the cable to stop the elevator over a distance of 3.4 m is -4356 N. The negative sign indicates that the tension is acting in the opposite direction of the weight of the elevator.

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Answer:

Three advantages of Reflecting Telescope over refracting Telescope:

1. there is no chromatic aberration in the Reflecting Telescope being mirror as an objective, while Refracting Telescope can suffer chromatic aberration being using lenses.

2. image of Reflecting Telescope is brighter due to large, polished curved mirrors than Refracting Telescope

3. Reflecting Telescope is compact and portable in size than Refracting Telescope.

Final answer:

Reflecting telescopes have three advantages over refracting telescopes: they gather more light, do not suffer from chromatic aberration, and are easier and less expensive to manufacture.

Explanation:

There are three advantages of reflecting telescopes over refracting telescopes:

To answer the question, we use principles from mechanics, involving the concepts of free fall and relative motion. The time taken by the bolt to hit the floor is around 0.78s and has different velocities to an observer inside and outside the elevator. According to a ground observer, the bolt travels a total distance of 4.95m.

To answer these questions, we need to involve the principles of physics, specifically mechanics dealing with motion. Firstly, since the bolt was initially at rest, and it falls under the influence of gravity, we use the formula for time in free fall t = √(2h/g), where h is the initial height (3.00 m) and g is the acceleration due to gravity (9.81 m/s²).

For (b), for an observer in the elevator, the bolt appears to fall straight down, so its velocity just as it hits the floor will be v = gt = 9.81*0.78 = 7.65 m/s.

For (c), for an observer on the floor landing, the elevator is moving upwards, thus the velocity of the bolt relative to the observer on the ground would be the sum of the falling velocity and the velocity of the elevator, so v = gt + elevator speed = 7.65 + 2.50 = 10.15 m/s.

For (d), the bolt would have traveled the height of the fall plus the distance the elevator traveled during the fall according to the ground observer. The distance the elevator moves is d = elevator speed * t = 2.50*0.78 = 1.95 m, therefore, the total distance the bolt travels is 3.00 m + 1.95 m = 4.95 m.

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Answer:

Explanation:

The first part of question is about the height of the rock from which it falls and hit the ground with speed of 11 m/s. Lets find out that height.

We will use the formula,

[tex]v^{2} _{f} = v^{2} _{i} + 2gh[/tex]

As the initial velocity of the rock was zero. [tex]v_{f} = 0[/tex]

[tex]v^{2} _{f} = 2gh\\ h = v^{2} _{f} / 2g\\h = \frac{(11 m/s)^{2} }{2(9.8 m/s^{2} )} \\h = 6.17 m[/tex]

Now we have to find the height from which the rock should be dropped and it's speed on hitting the ground should be 22 m/s.

Again we will use the same formula, same calculation but the value of velocity now should be 22 m/s.

[tex]v^{2} _{f} = v^{2} _{i} + 2gh[/tex]

[tex]v^{2} _{f} = 2gh\\ h = \frac{(22m/s)^{2} }{2(9.8 m/s^{2}) } \\h = 24.69 m[/tex]

To double the speed of the rock when it hits the ground (from 11 m/s to 22 m/s), the height from which it is dropped should be quadrupled. Hence, the rock should be dropped from a height of 24.5 meters.

The question is about finding the height from which a rock should be dropped so that its speed on hitting the ground is 22 m/s, given that when it is dropped from height h, its speed is 11 m/s. To solve this, we can use the physics equation for motion under constant acceleration, which is v² = 2gh, where v is the final velocity, g is the acceleration due to gravity, and h is the height of fall.

First, let us find the height h in the initial scenario: (11)² = 2*9.8*h => h = 6.125 m. Generally, the height h is proportional to the square of the speed, so if we double the final speed, the height should be quadrupled: h'(new height) = 4 * h = 4 * 6.125 m = 24.5 m.

Therefore, the rock should be dropped from a height of 24.5 m so that its speed on hitting the ground is 22 m/s.

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Answer:

a. 2.645 * 10^7 m/s^2

b. 2.645 * 10^4 N

Explanation:

Parameters given:

Velocity of rod = 2010m/s

Length of rod = 15.3cm = 0.153m

Mass of object placed at the end of the rod = 1g = 0.001kg

a. Centripetal acceleration is given as:

a = (v*v)/r

Where v = velocity

r = radius of curvature.

The radius of curvature in this case is equal to the length of the rod, since the rod makes the circular path of the motion.

Hence, centripetal acceleration at the end of the rod:

a = (2010*2010)/(0.153)

a = 26432156.86 m/s^2 = 2.64 * 10^7 m/s

b. The force needed to hold the object at the end of the rod is equal to the centripetal force at the end of the rod. Centripetal force is given as:

F = ma = (m*v*v)/r

Where a = centripetal acceleration

F = 0.001 * 2.64 * 10^7

F = 2.64 * 10^4N

The Centripetal acceleration at the end of rod is [tex]\bold {2.64 x 10^7\ m/s}[/tex] and Centripetal force required to hold the object is [tex]\bold { 2.64x 10^4\ N}[/tex].

Given  here:

Velocity of rod = 2010 m/s

Length of rod = 15.3cm = 0.153 m

Mass of object placed at the end of the rod = 1g = 0.001 kg

(A). Centripetal acceleration can be calculated by  

[tex]\bold {a =\dfrac { v^2}{r}}[/tex]

Where

v = velocity

r = radius of curvature.

Put the values,

[tex]\bold {a = \dfrac {(2010^2)}{(0.153)}}\\\\\bold {a = 26432156.86\ m/s^2 } \\\\\bold {a = 2.64 x 10^7\ m/s}[/tex]

(B).  Centripetal force can be calculated by

[tex]\bold {F = ma = \dfrac {(m v^2)}{r}}[/tex]

Where,

a = centripetal acceleration

Thus the force

[tex]\bold {F = 0.001 \times 2.64x10^7}\\\\\bold {F = 2.64x 10^4\ N}[/tex]

Therefore, the Centripetal acceleration at the end of rod is [tex]\bold {2.64 x 10^7\ m/s}[/tex] and Centripetal force required to hold the object is [tex]\bold { 2.64x 10^4\ N}[/tex].

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Radio waves have lower energy compared to other types of waves in the electromagnetic spectrum. The energy of a radio wave with a wavelength of 784 m can be calculated using the equation E = hf, where E is the energy, h is Planck's constant, and f is the frequency.

Radio waves fall within the electromagnetic spectrum which consists of various types of waves ranging from gamma rays to radio waves. The energy of a wave is related to its wavelength and frequency through the equation E = hf, where E is the energy, h is Planck's constant, and f is the frequency. Since radio waves have a longer wavelength, they have a lower frequency and therefore lower energy compared to other types of waves in the spectrum.

To calculate the energy of a radio wave with a wavelength of 784 m, we can use the equation c = λf, where c is the speed of light. Rearranging the equation to solve for f, we get f = c/λ. Plugging in the given wavelength of 784 m and the speed of light which is approximately 3 x 10^8 m/s, we can calculate the frequency as 3 x 10^8 m/s / 784 m = 3.83 x 10^5 Hz. Substitute this frequency value into the equation E = hf to calculate the energy of the radio wave.

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The energy of radio wave radiation for a 784-meter wavelength is approximately 2.53 x 10^-34 joules, calculated using Planck's equation for energy.

The energy E of radio wave radiation with a particular wavelength λ can be calculated using the equation:

E = hc/λ

Where:

E is the energy in Joules (J)

h is Planck's constant (6.626 x 10-34 J·s)

c is the speed of light in vacuum (about 3 x 108 m/s)

λ is the wavelength in meters (m)  

For a radio wave with a wavelength of 784 meters, we use these values to compute the energy:

E = (6.626 x 10-34 J·s) * (3 x 108 m/s) / 784 m

Calculating using the numerical values:

E = (6.626 x 10-34) * (3 x 108) / 784

E ≈ 2.53 x 10-34 J

Therefore, the energy of radio wave radiation with a wavelength of 784 meters is approximately 2.53 x 10-34 joules.

Explanation:

Formula to calculate resistivity is as follows.

            [tex]\rho = \frac{1}{qN \mu}[/tex]

                       = [tex]\frac{1}{(1.6 \times 10^{-19}) \times 2.72 \times 10^{17} \times 3900}[/tex] ohm/cm

                       = [tex]5.89 \times 10^{-3} ohm/cm[/tex]

As germanium is an intrinsic conductor. Hence, resistivity of Ge is as follows.

             [tex]\rho_{1} = \frac{1}{2qN_{o}\sqrt{\mu_{e}\mu_{r}}}[/tex]

                        = [tex]\frac{1}{2q(N_{A})^{\frac{1}{2}}\sqrt{\mu_{e}\mu_{r}}}[/tex]

                        =  [tex]\frac{1}{2 \times (1.6 \times 10^{-19}) \times (\sqrt{4.42 \times 10^{22}})\sqrt{(3900)(1900)}}[/tex] ohm/cm

                        = 0.546 [tex]ohm (\mu m)^{-1}[/tex]

Thus, we can conclude that resistivity of the Ge is 0.546 [tex]ohm (\mu m)^{-1}[/tex].

Answer:

1. d = 0.415 m.

2. Q = 2.285 x 10^{-10} C.

Explanation:

The electric field and potential can be found by the following equations:

[tex]E = \frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}\\V = \frac{1}{4\pi\epsilon_0}\frac{Q}{r}[/tex]

Applying these equations to the given variables yields

[tex]E = 12 = \frac{1}{4\pi\epsilon_0}\frac{Q}{d^2}\\V = 4.98 = \frac{1}{4\pi\epsilon_0}\frac{Q}{d}[/tex]

Divide the first line to the second line:

[tex]\frac{12}{4.98} = \frac{ \frac{1}{4\pi\epsilon_0}\frac{Q}{d^2}}{\frac{1}{4\pi\epsilon_0}\frac{Q}{d}}\\\frac{12}{4.98} = \frac{1}{d}\\d = 0.415~m[/tex]

Using this distance in either of the equations give the magnitude of the charge.

[tex]12 = \frac{1}{4\pi\epsilon_0}\frac{Q}{(0.415)^2}\\12 = \frac{1}{4\pi (8.8\times 10^{-12})}\frac{Q}{(0.415)^2}\\Q = 2.285 \times 10^{-10}~C[/tex]

Final answer:

The distance to the point charge is 0.415 meters. The magnitude of the charge is 231 picocoulombs (pC).

Explanation:

To calculate the distance to the point charge when the electric potential is 4.98 V and electric field magnitude is 12.0 V/m, we can use the relationship between electric field (E) and electric potential (V), which is E = -dV/dr. Rearranging the equation to solve for the distance (r), we get r = V/E = 4.98 V / 12.0 V/m = 0.415 m.

To find the magnitude of the charge (q), we can use the electric potential formula for a point charge: V = k*q/r, where k is the Coulomb's constant (8.99 x [tex]10^9 Nm^2/C^2[/tex]). Rearranging the formula and solving for q, we get q = V*r/k = (4.98 V * 0.415 m) / (8.99 x [tex]10^9 Nm^2/C^2[/tex]) = 2.31 x [tex]10^-^1^0[/tex] C or 231 pC.

The average vertical velocity of the rock thrown upwards from the height H to the ground is determined by the total vertical displacement divided by the total time, considering the downward acceleration due to gravity is negative.

The vertical component v_{av-y} of the average velocity of a small rock thrown straight up from the edge of a roof of height H to the ground can be found by using the kinematic equations of motion under constant acceleration. Since the acceleration due to gravity (g) is acting downward, it is represented as negative in the equations. Considering the upwards direction as positive and the downward acceleration as negative ensures that the final answer for v_{av-y} takes into account the total displacement, which includes both the upward and the downward path of the rock. The average velocity is thus the total displacement divided by the total time taken. The rock initially travels upwards to a maximum height before falling to the ground, completing its motion.

Explanation:

Relation between work and change in kinetic energy is as follows.

                 [tex]W_{net} = \Delta K[/tex]

Also,   [tex]\Delta K = K_{initial} - K_{final}[/tex]

                        = [tex](0 - 4.0 \times 10^{-17})[/tex] J

                        = [tex]-4.0 \times 10^{-17}[/tex] J

Let us assume that electric force on the electron has a magnitude F. The electron moves at a distance of 0.3 m opposite to the direction of the force so that work done is as follows.

                w = -Fd

       [tex]-4.0 \times 10^{-17} J = -F \times 0.3 m[/tex]

                F = [tex]1.33 \times 10^{-16}[/tex]  

Therefore, relation between electric field and force is as follows.

              E = [tex]\frac{F}{q}[/tex]

                 = [tex]\frac{1.33 \times 10^{-16}}{1.60 \times 10^{-19} C}[/tex]

                 = [tex]0.831 \times 10^{3}[/tex] C

Thus, we can conclude that magnitude of the electric field that will stop these electrons in a distance of 0.3 m is [tex]0.831 \times 10^{3}[/tex] C.

Answer:

Total work done in expansion will be [tex]3.60\times 10^5J[/tex]

Explanation:

We have given pressure P = 2.10 atm

We know that 1 atm [tex]=1.01\times 10^5Pa[/tex]

So 2.10 atm [tex]=2.10\times 1.01\times 10^5=2.121\times 10^5Pa[/tex]

Volume is increases from 3370 liter to 5.40 liter

So initial volume [tex]V_1=3.70liter[/tex]

And final volume [tex]V_2=5.40liter[/tex]

So change in volume [tex]dV=5.40-3.70=1.70liter[/tex]

For isobaric process work done is equal to [tex]W=PdV=2.121\times 10^5\times 1.70=3.60\times 10^5J[/tex]

So total work done in expansion will be [tex]3.60\times 10^5J[/tex]

Amount of heat absorbed: 214 J

Explanation:

When an object absorbs heat, its temperature increases according to the equation

[tex]Q=mC\Delta T[/tex]

where

Q is the heat absorbed

m is the mass of the object

C is the specific heat capacity of the material

[tex]\Delta T[/tex] is the change in temperature

For the iron skillet in this problem:

m = 28 g = 0.028 kg is the mass

[tex]C=450 J/kg^{\circ}C[/tex] is the iron specific heat capacity

[tex]\Delta T = 27-10=17^{\circ}C[/tex] is the increase in temperature

Solving for Q, we find the amount of heat absorbed:

[tex]Q=(0.028)(450)(17)=214 J[/tex]

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The electric field of a uniformly charged disk at different points on its axis was computed using the formula for the electric field due to a charged disk. a. 20.16 N/C b.  17.7 N/C c. 3.57 N/C d. 0.225 N/C

The student is asking for the calculation of the electric field at different points on the axis of a uniformly charged disk with a known charge density.

The formula to find the electric field E due to a charged disk along its axis at distance x from the center is given by E =((σ/2ε0) * (1 - (x/(sqrt(x^2 + r^2)))), where r is the radius of the disk and σ is the charge density. The constant ε0 is the permittivity of free space, and its value is approximately 8.85 x 10^-12 C^2/N·m^2.

a. For x=5.00 cm, E ~= 20.16 N/C

b. For x=10.0 cm, E ~= 17.7 N/C

c. For x=50.0 cm, E ~= 3.57 N/C

d. For x=200 cm, E ~= 0.225 N/C

These results reflect the fact that the strength of the electric field decreases as one moves farther away from the center of the disk.

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The electric fields at the respective distances are:

[tex]a. \( E(5.00 \, \text{cm}) = 4.369 \times 10^8 \, \text{N}/\text{C} \) b. \( E(10.0 \, \text{cm}) = 3.695 \times 10^8 \, \text{N}/\text{C} \) c. \( E(50.0 \, \text{cm}) = 9.187 \times 10^7 \, \text{N}/\text{C} \) d. \( E(200 \, \text{cm}) = 7.728 \times 10^6 \, \text{N}/\text{C} \)[/tex]

To calculate the electric field on the axis of a uniformly charged disk at a distance z from the center, we can use the formula derived from Gauss's law for a flat surface:

[tex]\[ E(z) = \frac{\sigma}{2\epsilon_0} \left( 1 - \frac{z}{\sqrt{z^2 + R^2}} \right) \][/tex]

where E(z)  is the electric field at a distance z from the center of the disk, [tex]\( \sigma \)[/tex] is the surface charge density, [tex]\( \epsilon_0 \)[/tex] is the vacuum permittivity [tex](\( 8.85 \times 10^{-12} \, \text{C}^2/(\text{N} \cdot \text{m}^2) \))[/tex], and R  is the radius of the disk.

Given:

- Radius of the disk, [tex]\( R = 35.0 \, \text{cm} = 0.35 \, \text{m} \)[/tex]

- Charge density, [tex]\( \sigma = 9.00 \times 10^{-3} \, \text{C}/\text{m}^2 \)[/tex]

- Vacuum permittivity,[tex]\( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/(\text{N} \cdot \text{m}^2) \)[/tex]

Now, let's calculate the electric field at the given distances:

a. At z = 5.00[tex]\text{cm} = 0.05 \, \text{m} \)[/tex]:

[tex]\[ E(0.05) = \frac{9.00 \times 10^{-3}}{2 \times 8.85 \times 10^{-12}} \left( 1 - \frac{0.05}{\sqrt{0.05^2 + 0.35^2}} \right) \] \[ E(0.05) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \left( 1 - \frac{0.05}{\sqrt{0.001225 + 0.1225}} \right) \][/tex]

[tex]\[ E(0.05) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \left( 1 - \frac{0.05}{0.3555} \right) \][/tex]

[tex]\[ E(0.05) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \left( 1 - 0.1406 \right) \] \[ E(0.05) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \times 0.8594 \] \[ E(0.05) = \frac{7.7346 \times 10^{-3}}{17.7 \times 10^{-12}} \] \[ E(0.05) = 4.369 \times 10^8 \, \text{N}/\text{C} \][/tex]

b. At z = 10.0[tex]\text{cm} = 0.1 \, \text{m} \)[/tex]:

[tex]\[ E(0.1) = \frac{9.00 \times 10^{-3}}{2 \times 8.85 \times 10^{-12}} \left( 1 - \frac{0.1}{\sqrt{0.1^2 + 0.35^2}} \right) \] \[ E(0.1) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \left( 1 - \frac{0.1}{\sqrt{0.01 + 0.1225}} \right) \] \[ E(0.1) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \left( 1 - \frac{0.1}{0.3674} \right) \][/tex]

[tex]\[ E(0.1) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \left( 1 - 0.2722 \right) \] \[ E(0.1) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \times 0.7278 \] \[ E(0.1) = \frac{6.5402 \times 10^{-3}}{17.7 \times 10^{-12}} \] \[ E(0.1) = 3.695 \times 10^8 \, \text{N}/\text{C} \][/tex]

c. At z = 50.0[tex]\text{cm} = 0.5 \, \text{m} \)[/tex]:

[tex]\[ E(0.5) = \frac{9.00 \times 10^{-3}}{2 \times 8.85 \times 10^{-12}} \left( 1 - \frac{0.5}{\sqrt{0.5^2 + 0.35^2}} \right) \] \[ E(0.5) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \left( 1 - \frac{0.5}{\sqrt{0.25 + 0.1225}} \right) \] \[ E(0.5) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \left( 1 - \frac{0.5}{0.6104} \right) \][/tex]

[tex]\[ E(0.5) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \left( 1 - 0.8193 \right) \] \[ E(0.5) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \times 0.1807 \] \[ E(0.5) = \frac{1.6263 \times 10^{-3}}{17.7 \times 10^{-12}} \] \[ E(0.5) = 9.187 \times 10^7 \, \text{N}/\text{C} \][/tex]

d. At z = 200 [tex]\text{cm} = 2.0 \, \text{m} \)[/tex]:

[tex]\[ E(2.0) = \frac{9.00 \times 10^{-3}}{2 \times 8.85 \times 10^{-12}} \left( 1 - \frac{2.0}{\sqrt{2.0^2 + 0.35^2}} \right) \] \[ E(2.0) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \left( 1 - \frac{2.0}{\sqrt{4 + 0.1225}} \right) \][/tex]

[tex]\[ E(2.0) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \left( 1 - \frac{2.0}{2.0317} \right) \] \[ E(2.0) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \left( 1 - 0.9848 \right) \][/tex]

[tex]\[ E(2.0) = \frac{9.00 \times 10^{-3}}{17.7 \times 10^{-12}} \times 0.0152 \] \[ E(2.0) = \frac{1.368 \times 10^{-4}}{17.7 \times 10^{-12}} \] \[ E(2.0) = 7.728 \times 10^6 \, \text{N}/\text{C} \][/tex]