A baseball has a mass of about 155 g. What is the magnitude of the momentum of a baseball thrown at a speed of 87 miles per hour? (Note that you need to convert mass to kilograms and speed to meters/second. A mile is 1.6 kilometers or 1600 meters.)

Answer:

-7 C

Explanation:

Assuming that object other than the styrofoam balls was part of the charge transfer. In order to maintain charge balance, the initial charge of the system must equal the ending charge. If all balls were uncharged initially, the ending charge on the third ball must be:

[tex]C_1+C_2+C_3 = 0\\3+4+C_3=0\\C_3=-7[/tex]

There must be -7 C of charge on the third ball.

Answer:

(a) Angular speed = 1344.8rev/min

(b) Centripetal acceleration = 8473.4m/s^2

(c) Force = 8.4734×10^-12N

(d) Ratio of force to bacterium's weight is 864.6

Explanation:

(a) Angular speed (w) = v/r

v = 60m/s, d = 0.850m, r = 0.850m/2 = 0.425m

w = 60/0.425 = 141.2rad/s = 141.2×9.524rev/min = 1344.8rev/min

(b) Centripetal acceleration = w^2r = 141.2^2 × 0.425 = 8473.4m/s^2

(c) Force = mass × centripetal acceleration = 1×10^-15 × 8473.4 = 8.4734×10^-12N

(d) Bacterium's weight = mass × acceleration due to gravity = 1×10^-15 × 9.8 = 9.8×10^-15N

Ratio of force to bacterium's weight = 8.4734×10^-12N/9.8×10^-15N = 864.6

The commercial jet's tires' rotation speed is calculated using their speed and diameter. The centripetal acceleration at the tire's edge is calculated using the speed and radius. The force with which a tiny bacterium clings to the rim is found using its mass and the centripetal acceleration.

We start by calculating the rotation speed of the tires. To do this, we use the known values of tire speed (60 m/s) and diameter (0.85 m). The first step is to convert speed into distance travelled per revolution, and diameter into circumference, then divide speed by circumference to get revolutions per second. We convert this into revolutions per minute for the answer to part (a). For part (b), we use the formula for centripetal acceleration, which involves the square of speed and the radius (half of the tire's diameter) to find acceleration at the rim. For part (c), we find the force exerted by the bacterium by multiplying its mass by the centripetal acceleration. Part (d) compares this force to the bacterium's weight, which is found by multiplying its mass by the acceleration due to gravity.

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The magnitude is used to quantify the size of the earthquakes (measures the energy released during the breakdown of a fault) while the intensity is a qualitative description of the effects of the earthquakes (it involves the perception of people as well as damage material and economic suffered due to the event). The relation between intensity and magnitude is

[tex]\frac{I_1}{I_2} =10^{(M1-M2)}[/tex]

Here,

M = Magnitude

I = Intensity of each one

Given

[tex]M_1 = 9[/tex]

[tex]M_2 =6[/tex]

Replacing,

[tex]\frac{I_1}{I_2} =10^{(9-6)}[/tex]

[tex]\frac{I_1}{I_2} = 10^3[/tex]

[tex]\frac{I_1}{I_2} = 1000[/tex]

So the aftershocks are 1000x weaker compared to the original earthquake. The correct answer is B.

In a helium balloon with a total mass of 160 amu containing helium-4 atoms, there would be 40 helium atoms, resulting in a total of 80 protons and 80 electrons.

The question involves calculating the number of protons and electrons in a helium balloon with a total mass of 160 amu, assuming it contains only 4He (helium-4). A helium-4 atom consists of 2 protons, 2 neutrons, and 2 electrons, with a total atomic mass of 4 amu. To find the number of helium atoms, we divide the total mass of helium in the balloon (160 amu) by the mass of a single helium-4 atom, which is 4 amu. This gives us 40 helium atoms.

Since each helium atom has 2 protons and 2 electrons, to find the total number of protons and electrons in 40 helium atoms, we multiply the number of helium atoms by the number of protons or electrons per atom. Therefore, there are 80 protons and 80 electrons in the helium balloon.

Answer:

1.5092 m

Explanation:

Time taken by the wave for the round trip is 8.8 ms

Velocity of sound at 20 degrees Celsius is 343 m/s

The distance to the object will be given by the one way trip

Time taken for one way trip is

[tex]\dfrac{8.8}{2}=4.4\ ms[/tex]

Distance is given by

[tex]Distance=Speed\times time\\\Rightarrow Distance=343\times 4.4\times 10^{-3}\\\Rightarrow Distance=1.5092\ m[/tex]

The distance between the motion sensor and the object is 1.5092 m

The distance between the motion sensor and the object is 1.5092 m

The motion sensor serves as a transmitter for pulses when it sends out pulses, and as a receiver, when it listens for echoes.

From the parameters given:

The distance traveled by the signal is:

d = s × Δ t

d = 343 m/s × 0.0088s

d = 3.0184 m

Provided that the distance is the circular trip between the motion sensor and the object,

Therefore, we can conclude that the distance between the motion sensor and the object will be equal to half the distance traveled by the signal.

i.e.

= 3.0184 m/2

= 1.5092  m

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Answer

Pressure, P = 1 atm

air density, ρ = 1.3 kg/m³

a) height of the atmosphere when the density is constant

   Pressure at sea level = 1 atm = 101300 Pa

   we know

   P = ρ g h

   [tex]h = \dfrac{P}{\rho\ g}[/tex]

   [tex]h = \dfrac{101300}{1.3\times 9.8}[/tex]

          h = 7951.33 m

height of the atmosphere will be equal to 7951.33 m

b) when air density decreased linearly to zero.

  at x = 0  air density = 0

  at x= h   ρ_l = ρ_sl

 assuming density is zero at x - distance

 [tex]\rho_x = \dfrac{\rho_{sl}}{h}\times x[/tex]

now, Pressure at depth x

[tex]dP = \rho_x g dx[/tex]

[tex]dP = \dfrac{\rho_{sl}}{h}\times x g dx[/tex]

integrating both side

[tex]P = g\dfrac{\rho_{sl}}{h}\times \int_0^h x dx[/tex]

[tex] P =\dfrac{\rho_{sl}\times g h}{2}[/tex]

 now,

[tex]h=\dfrac{2P}{\rho_{sl}\times g}[/tex]

[tex]h=\dfrac{2\times 101300}{1.3\times 9.8}[/tex]

  h = 15902.67 m

height of the atmosphere is equal to 15902.67 m.

Answer:

a) the number of antinodes increases , b) wavelength produced is constant.

, c) fundamental frequency is constant., d)  fundamental wavelength does not change, e)  the speed of the wave is constant

Explanation:

This is a resonance problem where we have a frequency generator, attached to a chain with a weight in its final part, at the two ends there is a node, point if movement. The condition for resonance of this system is

         λ = 2 L / n

  Where n is an integer

         L = n λ / 2

Let's review the problem questions.

A) As the length of the chain increases the number of wavelengths (Lam / 2) should increase, so the number of antinodes increases

B) when seeing the first equation the wavelength remains the same since the change in the length of the chain and the change in the number between are compensated, therefore for a configuration of generator frequency and weight applied the wavelength produced is constant.

C) the speed of the wave is

         v = λ f

In a string the speed is constant for a fixed applied weight, with the wavelength we did not change, therefore the fundamental frequency must also be constant.

D) The value of the fundamental wavelength does not change if the weight does not change, but there is a minimum chain length for this resonance to be observed and corresponds to n = 1

              L = λ / 2

E) the speed of the wave depends on the chain tension and its density if they do not change the speed of the wave does not change either

The frequency of wave B is half the frequency of wave A

Explanation:

The wavelength, the speed and the frequency of  a wave are related by the wave equation:

[tex]v=f \lambda[/tex]

where

v is the speed of the wave

f is its frequency

[tex]\lambda[/tex] is its wavelength

The equation can also be rewritten as

[tex]f=\frac{v}{\lambda}[/tex]

In this problem, we have wave A with wavelength [tex]\lambda_A[/tex] and speed v, so its frequency is

[tex]f_A=\frac{v}{\lambda_A}[/tex]

Then we have wave B, whose wavelength is twice that of wave A:

[tex]\lambda_B = 2 \lambda_A[/tex]

And its speed is the same; Therefore, its frequency is

[tex]f_B = \frac{v}{\lambda_B}=\frac{v}{2\lambda_A}=\frac{1}{2}(\frac{v}{\lambda_A})=\frac{f_A}{2}[/tex]

So, the frequency of wave B is half that of wave A.

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The Cartesian coordinates for the point are approximately: x ≈ 1.93185 m and, y ≈ 0.523599 m.

Here, we have to find the Cartesian coordinates (x, y) for a point given its distance from the origin (r) and the angle (θ) measured from the positive horizontal axis, we can use trigonometric functions.

In this case, r = 2 m and θ = 15°.

The Cartesian coordinates can be calculated as follows:

x = r * cos(θ)

y = r * sin(θ)

Given:

r = 2 m

θ = 15°

First, convert the angle from degrees to radians, since trigonometric functions in most programming languages use radians:

θ_rad = 15° * (π / 180) ≈ 0.261799 radians

Now, calculate the coordinates:

x = r * cos(θ_rad) = 2 m * cos(0.261799) ≈ 1.93185 m

y = r * sin(θ_rad) = 2 m * sin(0.261799) ≈ 0.523599 m

So, the Cartesian coordinates for the point are approximately:

x ≈ 1.93185 m

y ≈ 0.523599 m

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The Cartesian coordinates for the point with a distance of 2 m from the origin and an angle of 15° from the positive horizontal axis are approximately (1.93 m, 0.52 m), calculated using the cosine and sine of the angle.

To find the Cartesian coordinates for a point given its distance from the origin and the angle from the positive horizontal axis, you can use trigonometric functions. Since the distance from the origin is 2 m and the angle is 15°, you can calculate the x and y coordinates using the cosine and sine functions, respectively.

The x-coordinate is found by multiplying the distance by the cosine of the angle: x = 2 m × cos(15°). The y-coordinate is found by multiplying the distance by the sine of the angle: y = 2 m × sin(15°).

Using standard values for cos(15°) and sin(15°), the calculations would yield:

Therefore, the Cartesian coordinates are approximately (1.93 m, 0.52 m).

Answer:

Surface tension in water

Friction between tires and pavement

Dissolution of salt in water

Explanation:

Surface tension in water: It is due to the electrostatic force of attraction (cohesive force) between water molecules.

Friction between tires and pavement: It is due to the attractive force between tires and pavement.

Dissolution of salt in water: The ions of [tex]Na ^ +[/tex] and [tex]Cl ^ -[/tex] separate due to the strong attraction of water molecules.

Final answer:

Surface tension in water, friction between tires and pavement, and the dissolution of salt in water are phenomena due to electric interaction, which is rooted in the forces of cohesion, adhesion, and intermolecular attractions.

Explanation:

The phenomena that are due to electric interaction include surface tension in water, friction between tires and pavement, and the dissolution of salt in water. The electric interaction plays a crucial role in these phenomena through the forces of cohesion, adhesion, and intermolecular attractions, which are manifestations of the electromagnetic force. While the elliptical orbit of comets is primarily governed by gravitational forces and the binding of protons in an atomic nucleus by the strong nuclear force, surface tension, friction, and dissolution phenomena are deeply rooted in electrical interactions between atoms and molecules.

Answer:

  a = Np/10 yrs×[3.50^7 yrs /sec]

Explanation:

The energy of the single photon of frequency f or wave length λ  is given as

   E = hc / λ

since the glow warm emits energy 0.1 J/sec

that is  the number of photons n emitted by the photon per sec will be

n = 0.1 W / E

Thus, the number of photons emitted in 10 years

 N = n×3.15×10^7 sec/yr ×10 yr

Now, momentum associated with each photon

p= h / λ

and, momentum associated with N photon particles

P= N(h/λ)    

hence the change in the momentum of the glow is = Np in 10 years

Therefore, acceleration of the glow

                a = Np/10 yrs×[3.50^7 yrs /sec]

Answer:

The speed is 21.06 m/s.

Explanation:

Given that,

mass of glow worm = 5.0 g

Wavelength = 650 nm

Power = 0.10 W

Time = 10 years

The total energy emitted in a period [tex]\tau[/tex] is [tex]P\tau[/tex]

The energy of single photon of frequency or wavelength is

[tex]E=\dfrac{hc}{\lambda}[/tex]

The total number of photons emitted in a interval [tex]\tau[/tex] is then the total energy divided by the energy per photon.

[tex]N=\dfrac{P\tau}{E}[/tex]

[tex]N=\dfrac{P\tau\times\lambda}{hc}[/tex]

[tex]N=\dfrac{P\tau\times\lambda}{hc}[/tex]

We need to calculate the speed

Using de Broglie's relation applies to each photon and thus the total momentum imparted to the glow-worm

[tex]p=\dfrac{Nh}{\lambda}[/tex]

[tex]p=\dfrac{\dfrac{P\tau\times\lambda}{hc}\times h}{\lambda}[/tex]

[tex]p=\dfrac{P\tau}{c}[/tex]

[tex]v=\dfrac{P\tau}{mc}[/tex]

Put the value into the formula

[tex]v=\dfrac{0.10\times3.16\times10^{8}}{5.0\times10^{-3}\times3\times10^{8}}[/tex]

[tex]v=21.06\ m/s[/tex]

Hence, The speed is 21.06 m/s.

Answer:

1902.75 kg

Explanation:

From Law of conservation of momentum,

m₁u₁ + m₂u₂ = V (m₁ + m₂).................... Equation 1

make m₂ the subject of the equation,

m₂ = (m₁V - m₁u₁)/(u₂-V)..................... Equation 2

Where m₁ = mass of the truck, m₂ = mass of the car, u₁ initial velocity of the truck, u₂ = initial velocity of the car V = common velocity

Given: m₁ = 2537 kg, u₁ = 14, V= 8 m/s, u₂ = 0 m/s ( as the car was at rest waiting at a traffic light)

Substituting into equation 2.

m₂ =[2537(8) - 2537(14)]/(0-8)

m₂ = (20296-35518)/-8

m₂ = -15222/-8

m₂ = 1902.75 kg.

Thus the mass of the car = 1902.75 kg

Final answer:

The minimum possible temperature of the hot reservoir for an engine that does 25 J of work and exhausts 20 J of waste heat in each cycle, with a cold-reservoir temperature of 30°C is 408.87°C.

Explanation:

To find the minimum possible temperature of the hot reservoir for an engine that does 25 J of work and exhausts 20 J of waste heat during each cycle, with a cold-reservoir temperature of 30 °C, we have to use the concept of efficiency and the Carnot engine.

Firstly, let's convert the cold-reservoir temperature from Celsius to Kelvin:

Temperature in Kelvin (K) = Temperature in Celsius (°C) + 273.15

Tc (cold reservoir) = 30°C + 273.15 = 303.15 K

Next, we can calculate the efficiency (ε) of the engine using the formula:

ε = Work done (W) / Heat absorbed (Qh)

Qh (heat absorbed) = W (work done) + Qc (waste heat) = 25 J + 20 J = 45 J

So, ε = 25 J / 45 J = 0.555... (repeating)

The efficiency of a Carnot engine is also given by:

ε = 1 - (Tc/Th)

Now, we solve for Th (the hot reservoir temperature in Kelvin):

Th = Tc / (1 - ε)

Th = 303.15 K / (1 - 0.555...) = 303.15 K / 0.444... = 682.02 K

Finally, we convert the hot reservoir temperature back to Celsius:

Temperature in Celsius (°C) = Temperature in Kelvin (K) - 273.15

Th (hot reservoir) in °C = 682.02 K - 273.15 = 408.87°C

Thus, the minimum possible temperature of the hot reservoir is 408.87°C.

Answer:

[tex]49.6\times 10^5\ kg[/tex]

[tex]48.6\times 10^6\ N[/tex]

[tex]80.1\times 10^5\ N[/tex]

[tex]49.6\times 10^5\ kg[/tex]

Explanation:

[tex]1\ slug=14.5939\ kg[/tex]

[tex]340\times 10^3\ slug=340\times 10^3\times 14.5939=4961926\ kg[/tex]

The mass in SI unit is [tex]49.6\times 10^5\ kg[/tex]

Weight would be

[tex]W=mg\\\Rightarrow W=4961926\times 9.81\\\Rightarrow W=48676494.06\ N[/tex]

The weight in SI unit is [tex]48.6\times 10^6\ N[/tex]

[tex]1\ ft/s^2=0.3048\ m/s^2[/tex]

[tex]5.30\ ft/s^2=5.30\times 0.3048=1.61544\ m/s^2[/tex]

[tex]W=mg\\\Rightarrow W=4961926\times 1.61544\\\Rightarrow W=8015693.73744\ N[/tex]

The weight on the moon is [tex]80.1\times 10^5\ N[/tex]

The mass of an object is same anywhere in the universe.

So, the mass of the rocket on Moon is [tex]49.6\times 10^5\ kg[/tex]

According to the question,

The angle of incidence will be:

→ [tex]tan \Theta_1= \frac{l_1}{h_1}[/tex]

             [tex]= \frac{2.1}{1.3}[/tex]

             [tex]= 2.076[/tex]

then,

        [tex]\Theta_1 = 64.3^{\circ}[/tex]        

From air into water, we get

→       [tex]n_{air} Sin \Theta_1 = n_{water} Sin \Theta_2[/tex]

  [tex](1.00) Sin 64.3^{\circ} = (1.3) Sin \Theta_2[/tex]

                     [tex]\Theta_2 = 42.6^{\circ}[/tex]

hence,

The horizontal distance,

→ [tex]l_2 = l_1+h_2 tan \Theta[/tex]

By substituting the values, we get

      [tex]= 2.1+(2.1)tan 42.6[/tex]

      [tex]= 4.6 \ m[/tex]

Thus the above approach is right.

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To solve this problem we will define the data obtained in each of the sections. We know that the Net Force is equivalent to the Force in section 1, which can be found through mass flow and velocity, plus the force in section two, which can be found as the product between pressure and Area, so therefore we have

State 1:

[tex]\dot{m} = 200kg/s[/tex]

[tex]v_1 = 150m/s[/tex]

[tex]P_1 = 1600kPa[/tex]

State 2

[tex]v_2 = 2300m/s[/tex]

[tex]A_2 = 2.4m^2[/tex]

[tex]P_2 = 80kPa[/tex]

We have that net force is equal to

[tex]F_{net} = F_1 + F_2[/tex]

[tex]F_1 = \dot{m} (v_2-v_1)[/tex]

[tex]F_2 = (P_1-p_2)A_2[/tex]

Replacing,

[tex]F = \dot{m} (v_2-v_1) + (P_1-p_2)A_2[/tex]

[tex]F = 200(2300-150)+(1600-80)*10^3(2.4)[/tex]

[tex]F = 4078kN[/tex]

Therefore  the thrust force acting on the nozzle is 4078kN

The density of a fluid in which an object is submerged can be determined using the original mass of the object, the apparent submerged mass of the object, and the density of the object, using the formula provided above. Applying this formula to a 4.00-kg aluminum ball submersed in a liquid and appearing to be 2.10 kg yields a liquid density of 1247 kg/m^3.

According to Archimedes' Principle, the buoyant force on a submerged object is equal to the weight of the fluid displaced by the object. From this principle, if an object is fully submerged in a fluid then the volume of the fluid displaced is equal to the volume of the object.

To calculate the density of a liquid, using the mass of the object, the apparent mass, and the density of the object, the formula commonly used is:
ρliquid = (mobject - mapparent) / (mobject / ρobject - mapparent / ρobject).

Let's apply this formula to the situation you described. An aluminum ball of mass 4.00 kg is submerged in a liquid and has an apparent mass of 2.10 kg. The density of aluminum is approximately 2700 kg/m3. Therefore, the density of the liquid is (4.00kg - 2.10kg) / (4.00kg / 2700kg/m3 - 2.10kg / 2700kg/m3) = 1247 kg/m3.

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Archimedes' principle states that when a body is submerged in a liquid, it experiences an upward force equal to the weight of the liquid displaced by the object. The magnitude of the buoyant force is equal to the weight of the displaced liquid and is directed upward. The buoyant force can be used to determine the density of an object as well as the density of a liquid. The formula for Archimedes' principle is as follows:

Buoyant force = Weight of displaced fluid = density of fluid x volume of fluid displaced x g

where g is the acceleration due to gravity.

In this case, we can use the formula to determine the density of the liquid as follows:

Buoyant force = Weight of the object - Apparent weight of the object

Density of fluid x Volume of fluid displaced x g = m_object x g - m_apparent x g

Density of fluid = (m_object - m_apparent) / volume of fluid displaced

Now, substituting the given values, we get:

Density of fluid = (4.00 kg - 2.10 kg) / [(4.00 kg - 2.10 kg)/1000 kg/m³]

Density of fluid = 1900 kg/m³

Therefore, the density of the liquid is 1900 kg/m³.

Answer:

one forth.

Explanation:

car is moving at the speed of v

car stops, final speed = s

distance to stop the car = ?

using equation of motion

u_i² = u_f² + 2 a s

0² = v² - 2 a s

-ve sign is used because the car is decelerating.

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

now, if the velocity of the car is 2v distance to stop

[tex]s' = \dfrac{(2v)^2}{2a}[/tex]

[tex]s' = 4\dfrac{v^2}{2a}[/tex]

[tex]s' = 4 s[/tex]

[tex]s = \dfrac{s'}{4}[/tex]

now, the distance is one forth.

so, car with speed v has to cover one forth of the distance cover by car with speed 2 v.

Answer:

          ρ = 800 kg/m³

Explanation:

Let the volume of the sphere = V  

and the density of the sphere = ρ

density of water , ρ_w = 1000 Kg/m^3

tension in the string is one-fourth the weight of the sphere.

Tension in the rope ,

[tex]T = \rho V \dfrac{g}{4}[/tex]

for the sphere to in equilibrium

T + Weight = buoyant force

[tex]\rho V\dfrac{g}{4} + \rho V g = \rho_w V g[/tex]

[tex] \dfrac{\rho}{4}+\rho= \rho_w[/tex]

[tex] \dfrac{5\rho}{4}= 1000[/tex]

ρ = 800 kg/m³

density of the sphere is equal to 800 kg/m³

Final answer:

To calculate the density of the sphere tethered underwater, we apply Archimedes' principle and equilibrium of forces. Since the tension is one-fourth the sphere's weight, the sphere's density is found to be one-third more than the water's density.

Explanation:

To find the density of the sphere, let's consider the forces acting on the sphere when it is submerged in water. According to Archimedes' principle, the buoyant force on a submerged object is equal to the weight of the fluid that is displaced by the object.

The weight of the sphere (W) can be expressed as the product of its volume (V), its density (p_sphere), and the acceleration due to gravity (g). It can be written as W = V * p_sphere * g. Similarly, the weight of the water displaced by the sphere, which is also the buoyant force (B), is B = V * p_water * g, where p_water is the density of water.

Since the tension (T) in the string is one-fourth the weight of the sphere, we can write T = 1/4 * W. The sphere is in equilibrium, so the sum of the forces equals zero, leading to B = W - T. Substituting in the values, we get (V * p_water * g) = V * p_sphere * g - (1/4 * V * p_sphere * g). From this equation, we can solve for the density of the sphere p_sphere.

After simplifying, p_sphere = 4/3 * p_water, so the sphere's density is one-third more than the density of water.

Answer:

a)   I = √ (2G m³ (1/2r³ - 1/R)), b) I = √ (8 G m³ (1/2r -1/R))

Explanation:

.a) The relation of the Impulse and the moment is

                    I = Δp = m [tex]v_{f}[/tex] - m v₀

We can use Newton's second law with force the force of universal attraction

                  F = ma

                  G m m / r² = m a

                  dv / dt = G m / r²

Suppose re the direction where the spheres move is x

                 dv/dx  dx/dt = G m / x²

                  dv/dx  v = G m / x²

                  v dv = G m dx / x²

We integrate

                   v² / 2 = Gm (-1 / x)

We evaluate this integra from the lower limit v = 0 for x = R to the upper limit, where the spheres v = v and x = 2r are touched

                  v² / 2-0  = G M (-1 / R + 1 / 2r)

                  v = √ [2Gm (1 /2r - 1/ R) ]

The impulse on the sphere is

                 I = m vf - m v₀

                 I = m vf - 0

                 I = m √ (2Gm (1 / 2r-1 / R)

                 I = √ (2G m³ (1/2r³ - 1/R))

b) during the crash each sphere arrives with a velocity v and leaves with a velocity –v, the same magnitude but opposite direction

                      I = m [tex]v_{f}[/tex]- m v₀

                      I = m v - m (-v)

                      I = 2mv

                      I = 2m √ (2Gm (1 / 2r-1 / R)

                      I = √ (8 G m³ (1/2r -1/R))

We calculate the impulse received by each sphere before they make contact and during their elastic collision.

In this scenario, we have two identical hard spheres separated by a distance R. The spheres are released from rest and allowed to collide under the influence of their gravitational attraction. We need to find the magnitude of the impulse received by each sphere before they make contact and during their elastic collision.

(a) Before the spheres make contact, the magnitude of the impulse received by each sphere can be found using the equation:

Impulse = Change in momentum = Mass x Change in velocity.

Since both spheres are released from rest, their initial velocities are zero. Therefore, the change in velocity is the final velocity. Using the equation:

Final velocity = sqrt(2 x G x m / R).

Substituting the values, we can calculate the magnitude of the impulse received by each sphere before they make contact.

(b) During their elastic collision, the magnitude of the impulse received by each sphere can be found using the equation:

Impulse = Change in momentum = Mass x Change in velocity.

Since the spheres collide elastically, there is no change in velocity. Therefore, the magnitude of the impulse received by each sphere during their contact is zero.

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To solve this problem we will make a free body diagram to better understand the displacement measurements made by the body. From there we will apply the linear motion kinematic equations that describe the position of the body in reference to its vertical displacement, acceleration and speed. With this speed found we will apply the energy conservation theorem that will allow us to find the Force.

Equation of trajectory of a projectile is

[tex]y = xtan\theta - x^2 \frac{g}{2u^2cos^2\theta}[/tex]

Here

u = Initial velocity

x = Horizontal displacement

g = Acceleration due to gravity

y = Vertical displacement

We have that

[tex]x = 22m[/tex]

[tex]y = -2\sqrt{2m}[/tex]

Replacing we have that,

[tex]-2\sqrt{2} = 22tan45\° -\frac{9.8}{2u^2cos^2 45}(22)^2[/tex]

[tex]-2\sqrt{2} =22 -\frac{9.8*484}{2u^2(1/2)}[/tex]

[tex]-2\sqrt{2} =22 -\frac{4743.2}{u^2}[/tex]

[tex]u^2 = \frac{4743.2}{22+2\sqrt{2}}[/tex]

[tex]u = 191.03m/s[/tex]

From the work energy theorem

[tex]W_{net} = K_f +K_i[/tex]

Here,

[tex]K_f = \frac{1}{2} mu^2[/tex]

[tex]K_i = \frac{1}{2} m(0)^2 = 0[/tex]

[tex]W_{net} = W_m+W_g[/tex]

Where,

[tex]W_m =\text{Work by man} = F_s[/tex]

[tex]W_{g} = \text{Work by gravity} = -mgh[/tex]

Therefore

[tex]F_s -mgh = \frac{1}{2} mu^2[/tex]

[tex]F_s = \frac{m}{s} (\frac{u^2}{2}+gh)[/tex]

[tex]F_s = \frac{7.26}{2\sqrt{2}}(\frac{191.03}{2}+9.8*2)[/tex]

[tex]F_s = 295.477N[/tex]

Answer:

[tex]F=1.38*10^{-9}N[/tex]

Explanation:

According to Coulomb's law, the magnitude of the electric force between two point charges is directly proportional to the product of the magnitude of both charges and inversely proportional to the square of the distance that separates them:

[tex]F=\frac{kq_1q_2}{d^2}[/tex]

Here k is the Coulomb constant. In this case, we have [tex]q_1=-e[/tex], [tex]q_2=e[/tex] and [tex]d=4.09*10^-10m[/tex]. Replacing the values:

[tex]F=\frac{8.99*10^{9}\frac{N\cdot m^2}{C^2}(-1.6*10^{-19}C)(1.6*10^{-19}C)}{(4.09*10^{-10})^2}\\F=-1.38*10^{-9}N[/tex]

The negative sign indicates that it is an attractive force. So, the magnitude of the electric force is:

[tex]F=1.38*10^{-9}N[/tex]

Answer: FR=2.330kN

Explanation:

Write down x and y components.

Fx= FSin30°

Fy= FCos30°

Choose the forces acting up and right as positive.

∑(FR) =∑(Fx )

(FR) x= 5-Fsin30°= 5-0.5F

(FR) y= Fcos30°-4= 0.8660-F

Use Pythagoras theorem

F2R= √F2-11.93F+41

Differentiate both sides

2FRdFR/dF= 2F- 11.93

Set dFR/dF to 0

2F= 11.93

F= 5.964kN

Substitute value back into FR

FR= √F2(F square) - 11.93F + 41

FR=√(5.964)(5.964)-11.93(5.964)+41

FR= 2.330kN

The minimum force is 2.330kN

To determine the magnitude of force F to minimize the resultant FR of three forces, we can use the rule for finding the magnitude of a vector. By taking the square root of the sum of the squares of the components, we can find the magnitude of F.

The question is asking to determine the magnitude of force F so that the resultant FR of the three forces is as small as possible. To find the magnitude of the resultant force, we need to use the rule for finding the magnitude of a vector. We take the square root of the sum of the squares of the components. In this case, we have F = F₁ + F₂, and we can plug in the values given to find the magnitude of F.

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As a head-up, it is important to notice that a white dwarf only shines thanks to the stored energy and light, because a white dwarf doesn't have any hydrogen left to perform nuclear fusion.

Now the process:

First, the white dwarf accumulates all the extracted matter from its companion, onto its own surface. This extra matter increases the white dwarf's temperature and density.

After a while, the star reaches about 10 million K, so nuclear fusion can begin. The hydrogen that has been "stolen" from the other star and accumulated in the white dwarf's surface it's used for the fusion, dramatically increasing the star's brightness for a short time, causing what we know as a Nova.

As this fuel its quickly burnt out or blown into space, the star goes back to its natural white dwarf state. Since the white dwarf nor the companion star are destroyed in this process, it can happen countless of times during their lifespan.

The concept required to solve this problem is quantization of charge.

First the number of electrons will be calculated and then the total mass of the charge.

With these data it will be possible to calculate the percentage of load in the mass.

[tex]Q= ne[/tex]

Here Q is the charge, n is the number of electrons and e is the charge on the electron

[tex]n = \frac{Q}{e}[/tex]

Replacing,

[tex]n = \frac{4*10^{-6}C}{1.6*10^{-19}}[/tex]

[tex]n = 2.5 * 10^{13}77[/tex]

According to the quantization of charge the charge is defined as product of the number of electron and the charge on the electron

The total mass of the charge is

[tex]m= nm_e[/tex]

Here,

m = Mass of the charge

n = Number of electrons

[tex]m_e[/tex] = Mass of the electron

[tex]\text{Percentage change} = \frac{nm_e}{M}*100[/tex]

Replacing we have

[tex]\text{Percentage change} = \frac{(2.5*10^13)(9.1*10^{-28})}{33}*100[/tex]

[tex]\text{Percentage change} = 6.9*10^{-14} \%[/tex]

The percentage change in the mass of the 33 g comb during charging is extremely small, approximately [tex]\(6.91 \times 10^{-16}\%\),[/tex]  which is negligible.

To find the percentage change in mass of the comb due to charging, we need to understand the relationship between the amount of charge transferred and the mass of the electrons involved. Here's the step-by-step process:

1. Calculate the number of electrons corresponding to the given charge:

The charge of one electron[tex](\(e\))[/tex] is approximately[tex]\(1.6 \times 10^{-19}\)[/tex] coulombs.

Given:

  - Net charge [tex](\(Q\)) = 4.0 μC = \(4.0 \times 10^{-6}\) C[/tex]

The number of electrons n transferred can be calculated using the formula:

 [tex]\[ n = \frac{Q}{e} \][/tex]

[tex]\[ n = \frac{4.0 \times 10^{-6} \text{ C}}{1.6 \times 10^{-19} \text{ C/electron}} \][/tex]

 [tex]\[ n = 2.5 \times 10^{13} \text{ electrons} \][/tex]

2. Calculate the mass of the transferred electrons:

The mass of one electron [tex](\(m_e\))[/tex] is approximately [tex]\(9.11 \times 10^{-31}\)[/tex] kg.

The total mass[tex](\(m\))[/tex]  of the electrons transferred is:

[tex]\[ m = n \times m_e \][/tex]

[tex]\[ m = 2.5 \times 10^{13} \times 9.11 \times 10^{-31} \text{ kg} \][/tex]

 [tex]\[ m = 2.28 \times 10^{-17} \text{ kg} \][/tex]

3. Convert the mass of the comb to kilograms:

Given:

  - Mass of the comb = 33 g = 0.033 kg

4. Calculate the percentage change in mass:

The percentage change in mass is given by:

  [tex]\[ \text{Percentage change} = \left( \frac{\text{Change in mass}}{\text{Original mass}} \right) \times 100\% \][/tex]

[tex]\[ \text{Percentage change} = \left( \frac{2.28 \times 10^{-17} \text{ kg}}{0.033 \text{ kg}} \right) \times 100\% \][/tex]

[tex]\[ \text{Percentage change} = 6.91 \times 10^{-16} \% \][/tex]

Answer:

The Jupiter´s mass is approximately 1.89*10²⁷ kg.

Explanation:

The only force acting on Calisto while is rotating around Jupiter, is the gravitational force, as defined by the Newton´s Universal Law of Gravitation:

Fg = G*mc*mj / rcj²

where G = 6.67*10⁻¹¹ N*m²/kg², mc= Callisto´s mass, mj= Jupiter´s mass, and rcj = distance from Jupiter for Callisto= 1.88*10⁹ m.

At the same time, there exists a force that keeps Callisto in orbit, which is the centripetal force, that actually is the same gravitational force we have already mentioned.

This centripetal force is related with the period of the orbit, as follows:

Fc = mc*(2*π/T)²*rcj.

In order to be consistent in terms of units, we need to convert the orbital period, from days to seconds, as follows:

T = 16.69 days* 86,400 (sec/day) = 1.44*10⁶ sec.

We have already said that Fg= Fc, so we can write the following equality:

G*mc*mj / rcj² = mc*(2*π/T)²*rcj

Simplifying common terms, and solving for mj, we get:

mj = 4*π²*(1.88*10⁹)³m³ / ((1.44*10⁶)² m²*6.67*10⁻11 N*m²/kg²)

mj = 1.89*10²⁷ kg.

Answer: Mass of Jupiter ~= 1.89 × 10^23 kg

Explanation:

Given:

Period P= 16.69days × 86400s/day= 1442016s

Radius of orbit a = 1.88×10^6km × 1000m/km

r = 1.88 × 10^9 m

Gravitational constant G= 6.67×10^-11 m^3 kg^-1 s^-2

Applying Kepler's third law, which is stated mathematically as;

P^2 = (4π^2a^3)/G(M1+M2) .....1

Where M1 and M2 are the radius of Jupiter and callisto respectively.

Since M1 >> M1

M1+M2 ~= M1

Equation 1 becomes;

P^2 = (4π^2a^3)/G(M1)

M1 = (4π^2a^3)/GP^2 .....3

Substituting the values into equation 3 above

M1 = (4 × π^2 × (1.88 × 10^9)^3)/(6.67×10^-11 × 1442016^2)

M1 = 1.89 × 10^27 kg

Answer:

- Contact 1 with 3 ,  initial charge of 1.8 C.

- contact 1 with 2 and then 1 with 3 , first body should have 3.6 C

Explanation:

The excess charge on a body is distributed evenly throughout the body.

We can have two different configurations:

- Contact 1 with 3

When the third body was touched with the first, the initial charge was distributed between the two, so that when each one separated, it had half the charge, in this configuration the first body should have an initial charge of 1.8 C.

- contact 1 with 2 and then 1 with 3

Another possible configuration of the exercise is that the first body touches the second and the charge decrease to the half and then touches the third where it again decreases by half, so that the first body only gives it every ¼ of its initial load.

Therefore in this configuration if the third body has a load of 0.9C the first body should have 3.6 C

Answer:

c. No. An equation may have consistent units but still be numerically invaid.

Explanation:

For an equation to be corrected, it should have consistent units and also be numerically correct.

Most equation are of the form;

(Actual quantity) = (dimensionless constant) × (dimensionally correct quantity)

From the above, without the dimensionless constant the equation would be numerically wrong.

For example; Kinetic energy equation.

KE = 0.5(mv^2)

Without the dimensionless constant '0.5' the equation would be dimensionally correct but numerically wrong.

Answer:

No. An equation may have consistent units but still be numerically invalid.

Explanation:

Answer: 21.48m

Explanation:

Using the inverse square law, the intensity of sound is inversely proportional to the square of the distance.

Intensity ∝ 1/distance²

let l represent intensity and d represent distanc

l₁/l₂ = d₂²/d₁² ......1

given;

l₁ = 75dB

l₂ = 65dB

d₁ = 20 m

substituting into eqn 1, we have;

75/65 = d₂²/20²

d₂² = 75 × 20²/65

d₂² = 461.54

d₂ = √461.54

d₂ = 21.48m