A heat engine operates at 30% of its maximum possible efficiency and needs to do 995 J of work. Its cold reservoir is at 22 ºC and its hot reservoir is at 610 ºC. (a) How much energy does it need to extract from the hot reservoir? (b) How much energy does it deposit in the cold reservoir?

The angular displacement of the solid disk after 5 seconds is 0 rad.

To determine the angular displacement of the solid disk after 5 seconds, we can use the formula:

Δθ = ΔErot / (I * ω)

where Δθ is the angular displacement, ΔErot is the change in rotational kinetic energy, I is the moment of inertia of the disk, and ω is the angular velocity of the disk.

The moment of inertia of a solid disk rotating around an axis through its center perpendicular to its plane is given by:

I = (1/2) * m * r2

where m is the mass of the disk and r is the radius.

Given that ΔErot = 20 J/s, m = 4 kg, r = 2 m, and the disk starts from rest, we can calculate the angular displacement:

Δθ = ΔErot / (I * ω) = 20 / [(1/2) * 4 * 22 * ω]

Since the disk starts from rest, the initial angular velocity ω is 0. Therefore, the angular displacement after 5 seconds is:

Δθ = 20 / [(1/2) * 4 * 22 * 0] = 0 rad

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The work done by the electric force on the moving point charge is approximately -5.09 × 10^-5 J.

Work done by the electric force is given by the equation W = q1 * q2 * (1/r1 - 1/r2), where q1 and q2 are the charges, r1 is the initial distance, and r2 is the final distance.

In this case, q1 = 2.30 μC, q2 = -5.00 μC, r1 = 0.170 m, and r2 = 0.250 m. Plugging these values into the equation and solving for W, we get:

W = (2.30 μC) * (-5.00 μC) * [1/√(0.170^2) - 1/√(0.250^2 + 0.250^2)]

After simplifying, the work done is approximately -5.09 × 10^-5 J.

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Answer:

[tex]1.15669\times 10^{-7}\ C[/tex]

Explanation:

[tex]\theta[/tex] = Angle with which the electric field is hung = 23°

m = Mass of ball = 5 g

E = Electric field = [tex]1.8\times 10^5\ N/C[/tex]

T = Tension

q = Charge

We have the equations

[tex]Tcos\theta=mg[/tex]

[tex]Tsin\theta=qE[/tex]

Dividing the equations

[tex]tan\theta=\dfrac{mg}{qE}\\\Rightarrow q=\dfrac{mgtan\theta}{E}\\\Rightarrow q=\dfrac{5\times 10^{-3}\times 9.81\times tan23}{1.8\times 10^5}\\\Rightarrow q=1.15669\times 10^{-7}\ C[/tex]

The magnitude of the charge is [tex]1.15669\times 10^{-7}\ C[/tex]

Answer:

It will go 96 times around the track.

Explanation:

Given that,

Distance covered by the race car, d = 765 km

Radius of the circular sprint track, r = 1.263 km

Let n times did it go around the track. It is given by :

[tex]n=\dfrac{d}{C}[/tex]

C is the circumference of the circular path, [tex]C=2\pi r[/tex]

[tex]n=\dfrac{d}{2\pi r}[/tex]

[tex]n=\dfrac{765}{2\pi \times 1.263}[/tex]

[tex]n=96.4[/tex]

Approximately, n = 96

So, it will go 96 times around the track. Hence, this is the required solution.

With a track radius of 1.263 km, the car completes approximately 96.50 laps.

The formula for the circumference (C) of a circle is: C = 2πr, where r is the radius of the circle, and π (pi) is approximately 3.14159. Using the given radius of 1.263 km, we can calculate the circumference of the track:

C = 2π(1.263 km) ≈ 2(3.14159)(1.263 km) ≈ 7.932 km (rounded to three decimal places).

Divide the total distance traveled by the circumference of the track:

Number of laps = Total distance traveled ÷ Circumference of the track
Number of laps = 765 km ÷ 7.932 km ≈ 96.50 laps.

Therefore, the race car would have completed approximately 96.50 laps around the track.

Answer:

(a) 5.6 m/s, 7.2 m/s, and 8.8 m/s, respectively.

(b) 0.8 m/s^2

(c) 4.8 m/s

(d) 6 s

(e) 5.2 m

Explanation:

(a) The average velocity is equal to the total displacement divided by total time.

For the first 2s. interval:

[tex]V_{\rm avg} = \frac{\Delta x }{\Delta t} = \frac{25.6 - 14.4}{2} = 5.6~{\rm m/s}[/tex]

For the second 2s. interval:

[tex]V_{\rm avg} = \frac{40 - 25.6}{2} = 7.2~{\rm m/s}[/tex]

For the third 2s. interval:

[tex]V_{\rm avg} = \frac{57.6 - 40}{2} = 8.8~{\rm m/s}[/tex]

(b) Every 2 s. the velocity increases 1.6 m/s. Therefore, for each second the velocity increases 0.8 m/s. So, the acceleration is 0.8 m/s2.

(c) The sled starts from rest with an acceleration of 0.8 m/s2.

[tex]v^2 = v_0^2 + 2ax\\v^2 = 0 + 2(0.8)(14.4)\\v = 4.8~{\rm m/s}[/tex]

(d) The following kinematics equation will yield the time:

[tex]\Delta x = v_0 t + \frac{1}{2}at^2\\14.4 = 0 + \frac{1}{2}(0.8)t^2\\t = 6~{\rm s}[/tex]

(e) The same kinematics equation will yield the displacement:

[tex]\Delta x = v_0t + \frac{1}{2}at^2\\\Delta x = (4.8)(1) + \frac{1}{2}(0.8)1^2\\\Delta x = 5.2~{\rm m}[/tex]

Answer:

On moon time period will become 2.45 times of the time period on earth

Explanation:

Time period of simple pendulum is equal to [tex]T=2\pi \sqrt{\frac{l}{g}}[/tex] ....eqn 1 here l is length of the pendulum and g is acceleration due to gravity on earth

As when we go to moon, acceleration due to gravity on moon is [tex]\frac{1}{6}[/tex] times os acceleration due to gravity on earth

So time period of pendulum on moon is equal to

[tex]T_{moon}=2\pi \sqrt{\frac{l}{\frac{g}{6}}}=2\pi \sqrt{\frac{6l}{g}}[/tex] --------eqn 2

Dividing eqn 2 by eqn 1

[tex]\frac{T_{moon}}{T}=\sqrt{\frac{6l}{g}\times \frac{g}{l}}[/tex]

[tex]T_{moon}=\sqrt{6}T=2.45T[/tex]

So on moon time period will become 2.45 times of the time period on earth

Final answer:

The period of a pendulum on the Moon would be longer because the Moon's gravity is weaker. To achieve the same one-second period as on Earth, a pendulum needs to be much shorter due to the Moon's 1/6th gravitational acceleration. Consequently, a pendulum's frequency would decrease if taken from Earth to the Moon.

Explanation:

The period of a simple pendulum is affected by the acceleration due to gravity, which is less on the Moon than on Earth. Hence, if you took a pendulum clock, like a grandfather clock, to the Moon, its pendulum would swing more slowly because of the Moon's weaker gravity. To maintain a steady tick-tock of one second per period on the Moon, the pendulum would need to be much shorter. A grandfather clock pendulum designed to have a two-second period on Earth with a length of 50 cm would need to be only 8.2 cm long on the Moon to achieve the same period, since the Moon's gravity is 1/6th that of Earth. Therefore, if a pendulum from Earth was taken to the Moon, its frequency would decrease because the acceleration due to gravity on the Moon is less than that on Earth.

Answer:

D) The potential difference between the plates increases.

Explanation:

The capacitance of a parallel plate capacitor having plate area A and plate separation d is C=ϵ0A/d.  

Where ϵ0 is the permittivity of free space.  

A capacitor with increased distance, will have a new capacitance C1=ϵ0kA/d1

Where d1 = nd  

since d1 > d

therefore n >1

n is a factor derived as a result of the increased distance

Therefore the new capacitance becomes:

              C1=ϵ0A/d1

        C1= ϵ0A/nd

        C1= C/n  -------1

Where C1 is the capacitance with increased distance.

This implies that the charge storing capacity of the capacitor with increased plate separation decreases by a factor of (1/n) compared to  that of the capacitor with original distance.

Given points

The charge stored in the original capacitor Q=CV

The charge stored in the original capacitor after inserting dielectric  Q1=C1V1

The law of conservation of energy states that the energy stored is constant:

i.e Charge stored in the original capacitor is same as charge stored after the dielectric is inserted.

Charge before plate separation increase same as after plate separation increase

Q   = Q1

CV = C1V1

  CV = C1V1  -------2

We derived C1=C/n in equation 1. Inserting this into equation 2

   CV = (CV1)/n

   V1 = n(CV)/C

        = n V

Since n > 1 as a result of the derived new distance, the new voltage will increase

Relation Between Velocity And Wavelength

Wavelength is the measure of the length of a complete wave cycle. The velocity of a wave is the distance travelled by a point on the wave. In general, for any wave the relation between Velocity and Wavelength is proportionate. It is expressed through the wave velocity formula.

Velocity And Wavelength

For any given wave, the product of wavelength and frequency gives the velocity. It is mathematically given by wave velocity formula written as-

V=f×λ

Where,

V is the velocity of the wave measure using m/s.

f is the frequency of the wave measured using Hz.

λ is the wavelength of the wave measured using m.

Velocity and Wavelength Relationtion

Amplitude, Frequency, wavelength, and velocity are the characteristic of a wave. For a constant frequency, the wavelength is directly proportional to velocity.

Given by:

V∝λ

Example:

For a constant frequency, If the wavelength is doubled. The velocity of the wave will also double.

For a constant frequency, If the wavelength is made four times. The velocity of the wave will also be increased by four times.

Hope you understood the relation between wavelength and velocity of a wave. You may also want to check out these topics given below!

Relation between phase difference and path difference

Relation Between Frequency And Velocity

Relation Between Escape Velocity And Orbital Velocity

Relation Between Group Velocity And Phase Velocity

The relationship between wavelength, wave frequency, and wave velocity is described by the equation v = fλ. Wavelength and frequency are inversely proportional given a constant wave velocity – high frequency correlates with short wavelength and vice versa.

In Physics, there's a mathematical relationship between wavelength, wave frequency, and wave velocity for any type of wave motion. This relationship is often stated as v = fλ, where v is wave velocity, f is the frequency of the wave, and λ is the wavelength. The wavelength is the distance between identical parts of the wave, while the velocity is the speed at which the disturbance moves, and the frequency is the rate of oscillation of the wave.

When you look at this formula, it becomes clear that if the wave velocity (v) is constant, a wave with a longer wavelength (λ) will have lower frequency (f). On the other side, higher frequency means shorter wavelength. This is because frequency and wavelength are inversely proportional in the given formula.

For instance, the speed of light in vacuum is a constant value (approximately 3.00×108 m/s). So, if a certain light wave has a larger wavelength, its frequency will be lower to ensure this speed remains consistent.

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Answer:

0.0473m

Explanation:

345 ml = 0.000354 m3

6.5 cm = 0.065 m

20g = 0.02 kg

Since can is half filled with water, the water volume is 0.000354 / 2 = 0.000177 m cubed

Let water density be 1000kg/m3, the mass of this half-filled water is

1000*0.000177 = 0.177 kg

The total water-can system mass is 0.177 + 0.02 = 0.197 kg

For the system to stay balanced, this mass would be equal to the mass of the water displaced by the can submerged

The volume of water displaced, or submerged can is

0.197 / 1000 = 0.000197 m cubed

Then the volume of the can that is not submerged, aka above water level is

0.000354 - 0.000197 = 0.000157 m cubed

The base area of the can is

[tex]A = \pi r^2 = \pi (d/2)^2 = \pi (0.065)^2 = 0.003318 m squared[/tex]

The length of the can that is above water is

0.000157 / 0.003318 = 0.0473 m

The half-filled soda can displaces the equivalence of its own weight in water when placed in it. Half of the soda can's total volume will always be submerged since it is only half-filled i.e., half of the can's mass is displacing water.

The subject of this problem is the principles of buoyancy and volume. A half-filled soda can placed in water will displace its own weight of the water. The length of the can above the water level can be calculated using an understanding of volume and displacement.

First, calculate the volume of the can using the formula for the volume of a cylinder V = πr²h, where r is radius and h is height. Given the diameter of the can is 6.5 cm, the radius is 3.25 cm. The height can be calculated by rearranging the volume formula to find h. We know that the can's complete volume is 345 mL, so h (full can height) = V / (πr²).

From this, we can calculate the height of the can that is submerged in water. Since the can is half-filled, it displaces half its full weight in water. So half of the can's total volume will always be submerged. Therefore, the length of the can above the water will be half the total height of the can.

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Answer:

9. The force is a force of attraction and it is 2.95N

10. The magnitude of acceleration 35.12m/s^2 and the direction of this acceleration is away from the other balloon.

Explanation:

Parameters given:

Q1 = 3.4 * 10^-6C

Q2 = - 5.1 * 10^-6C

Distance between the two balloons = 23cm = 0.23m

9. Force acting between the two balloons is a force of attraction because they are unlike charges. Hence, the force between them is:

F = kQ1Q2/r^2

F = (9 *10^9 * 3.4 * 10^-6 * -5.1 * 10^-6)/(2.3 * 10^-1)^2

F = (1.56 * 10^-1)/(5.29 * 10^-2)

F = - 2.95N

10. Assuming that Balloon A has a mass, m, of 0.084kg, then:

F = ma

Where a = acceleration

a = F/m

a = -2.95/0.084

a = - 35.12m/s^2

The acceleration has a magnitude of 35.12m/s^2 and its direction is away from balloon B.

The negative sign shows that the balloon A is slowing down as it moves towards balloon B. Hence, it's velocity is reducing slowly.

Answer:

22.8077659955 m deep

Explanation:

t = Time taken

u = Initial velocity

v = Final velocity

s = Displacement

a = Acceleration

g = Acceleration due to gravity = 9.81 m/s² = a

[tex]v^2-u^2=2as\\\Rightarrow v=\sqrt{2as+u^2}\\\Rightarrow v=\sqrt{2\times 9.81\times 6.1+0^2}\\\Rightarrow v=10.9399268736\ m/s[/tex]

[tex]v=u+at\\\Rightarrow t=\dfrac{v-u}{a}\\\Rightarrow t=\dfrac{10.9399268736-0}{9.81}\\\Rightarrow t=1.11518112881\ s[/tex]

Time taken to fall through the foam

[tex]3.2-1.11518112881=2.08481887119\ s[/tex]

Distance is given by

[tex]s=vt\\\Rightarrow s=10.9399268736\times 2.08481887119\\\Rightarrow s=22.8077659955\ m[/tex]

The vat is 22.8077659955 m deep

The depth of the vat obtained is 44.076 m

H = ½gt²

H = ½ × 9.8 × 3.2²

H = 4.9 × 10.24

H = 50.176 m

Depth of vat = H – h

Depth of vat = 50.176 – 6.10

Depth of vat = 44.076 m

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Answer:

The force of gravitational attraction increases by 9 as the two point masses increase by 3.

Explanation:

Gravitational force of attraction, F is the force that pulls two point masses, m and M which are separated by a distance, d.

Mathematically,

Fg = GMm/r^2

Initially,

M1 = M1

M2 = M2

The remaining parameters are unchanged.

Fg1 = G * M1 * m1/(d/2)^2

Then,

M1 = 3M1

M2 = 3M2

Fg2 = G * 3M1 * 3M2/(d/2)^2

Making the constants G/(d/2)^2 the subject of formula and then comparing both equations,

= Fg1 = (M1 * M2); Fg2 = (9 * M1 * M2)

= Fg2 = 9 * Fg1

The force of gravitational attraction increases by 9 as the two point masses increase by 3.

Answer:

Cu = 1453.72J/Kg°C

The heat capacity of the liquid is 1453.72J/Kg°C

Explanation:

At equilibrium, assuming no heat loss to the surrounding we can say that;

Heat gained by ice + heat gained by cold water = heat loss by hot unknown liquid (24°C) + heat loss by aluminium calorimeter.

Given;

Mass of ice = mass of cold water = mc = 30g = 0.03kg

Mass of hot unknown liquid mh= 300g = 0.3kg

Mass of aluminium calorimeter ma= 100g = 0.1 kg

change in temperature cold ∆Tc = (4-0) = 4°C

Change in temperature hot ∆Th = 24-4 = 20°C

Specific heat capacity of water Cw= 4186J/Kg°C

Specific heat capacity of aluminium Ca = 900J/kg°C

Specific heat capacity of unknown liquid Cu =?

Heat of condensation of ice Li = 334000J/Kg

So, the statement above can be written as.

mcLi + mcCw∆Tc = maCa∆Th + mhCu∆Th

Making Cu the subject of formula, we have;

Cu = [mcLi + mcCw∆Tc - maCa∆Th]/mh∆Th

Substituting the values we have;

Cu = (0.03×334000 + 0.03×4186×4 - 0.1×900×20)/(0.3×20)

Cu = 1453.72J/Kg°C

the heat capacity of the liquid is 1453.72J/Kg°C

The amount of current that flows through this given portion of a cell membrane, calculated using Ohm's law and the properties of the membrane, is 1.60 µA. If the side dimensions of the membrane are halved, the current will decrease by a factor of 4.

The relevant concept needed to answer these questions is Ohm's Law, defined as Voltage = Current x Resistance. In this context, Resistance = Resistivity x (Thickness/Area) and the area is a square.

First, calculate the resistance: R = ρ x (Thickness/ Area)
Remove the micrometers units of the area and convert it into meters to match the ρ units. So, you get an area of 1.3 x 10^-6 m x 1.3 x 10^-6 m = 1.69 x 10^-12 m^2. Then, R = 1.30 x 10^7 Ω*m x (7.50 x 10^-9 m / 1.69 x 10^-12 m^2) = 57.404 Ω.

By plugging the calculated resistance and given voltage into Ohm's Law, we can find the current: I = V/R = 92.2 x 10^-3 V / 57.4 Ω = 1.60 μA

If the side dimensions are halved, the area of the membrane becomes one-fourth of the original, thus the resistance increases by a factor of 4. According to Ohm's Law, as resistance increases, the current decreases, meaning that if the resistance is multiplied by 4, the current will decrease by a factor of 4.

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a) Therefore, the amount of current that flows through this portion of the membrane is approximately [tex]\({1.60 \times 10^{-6} \, \text{A}} \)[/tex]. b) The correct answer is decrease by a factor of 5208. The current decreases by a factor of approximately 5208.

Part (a): Determine the amount of current that flows through this portion of the membrane

To find the current ( I ) flowing through the membrane portion, we use Ohm's law and the given potential difference ( V ) across the membrane.

1. Calculate the resistance ( R ) of the membrane:

The resistivity [tex](\( \rho \))[/tex] is given as [tex]\( 1.30 \times 10^7 \) ohms\·m.[/tex]

First, calculate the cross-sectional area ( A ) of the membrane portion:

[tex]\[ A = 1.3 \, \mu \text{m} \times 1.3 \, \mu \text{m} = (1.3 \times 10^{-6} \, \text{m})^2 = 1.69 \times 10^{-12} \, \text{m}^2 \][/tex]

Then, calculate the resistance ( R ):

[tex]\[ R = \frac{\rho \cdot L}{A} \][/tex]

[tex]\[ R = \frac{1.30 \times 10^7 \, \text{ohm} \cdot \text{m} \cdot 7.50 \times 10^{-9} \, \text{m}}{1.69 \times 10^{-12} \, \text{m}^2} \][/tex]

[tex]\[ R = \frac{9.75 \times 10^{-2}}{1.69 \times 10^{-12}} \approx 5.77 \times 10^7 \, \text{ohms} \][/tex]

2. Calculate the current ( I ):

Ohm's law states [tex]\( I = \frac{V}{R} \).[/tex]

Given potential difference [tex]\( V = 92.2 \, \text{mV} = 92.2 \times 10^{-3} \, \text{V} \):[/tex]

[tex]\[ I = \frac{92.2 \times 10^{-3} \, \text{V}}{5.77 \times 10^7 \, \text{ohms}} \approx 1.60 \times 10^{-6} \, \text{A} \][/tex]

Part (b): By what factor does the current change if the side dimensions of the membrane portion is halved?

If the side dimensions of the membrane portion are halved, the cross-sectional area ( A ) of the membrane will decrease by a factor of ( 4 ) (since both length and width are halved).

1. New cross-sectional area ( A' ):

[tex]\[ A' = \left( \frac{1.3 \, \mu \text{m}}{2} \right) \times \left( \frac{1.3 \, \mu \text{m}}{2} \right) = \left( \frac{1.3}{2} \times 10^{-6} \, \text{m} \right)^2 = 0.325 \times 10^{-12} \, \text{m}^2 \][/tex]

2. New resistance ( R' ):

Using the same resistivity [tex]\( \rho \)[/tex] and thickness ( L ):

[tex]\[ R' = \frac{\rho \cdot L}{A'} = \frac{1.30 \times 10^7 \cdot 7.50 \times 10^{-9}}{0.325 \times 10^{-12}} \approx 3.00 \times 10^8 \, \text{ohms} \][/tex]

3. New current ( I' ):

[tex]\[ I' = \frac{V}{R'} = \frac{92.2 \times 10^{-3}}{3.00 \times 10^8} \approx 3.07 \times 10^{-10} \, \text{A} \][/tex]

4. Calculate the factor by which the current changes:

[tex]\[ \frac{I'}{I} = \frac{3.07 \times 10^{-10}}{1.60 \times 10^{-6}} \approx 1.92 \times 10^{-4} \][/tex]

Since the current decreases, we consider the reciprocal:

[tex]\[ \frac{I}{I'} \approx \frac{1}{1.92 \times 10^{-4}} \approx 5208 \][/tex]

This process occurs because there is a contact between the carpet and the person's feet. Basically that contact generates the transfer of some electrons to the carpet on dry winter days.

In this way a person is charged when dragging bare feet on the carpet on a dry winter day.

Therefore, the net positive charge occurs on the surface of the carpet.

A person becomes charged when shuffling across a carpet due to the transfer of electrons from the feet to the carpet, leaving a net positive charge. The lack of humidity on a dry winter day allows the static charge to build up, leading to noticeable static shocks when touching a metal object. Humidity helps in dissipating the charge, making shocks less common on humid days.

When a person shuffles across a carpet with bare feet, they can become "charged" through a process known as charging by friction. This occurs when electrons are transferred from one surface to another due to the contact and relative motion between them. In this case, electrons move from the person's feet to the carpet, leaving the feet with a net positive charge.

Materials have different affinities for electrons, and when they come into close contact, the one with the higher affinity will take on electrons from the other. Since a dry winter day has low humidity, there is less moisture in the air to carry away the excess electrons. Therefore, the static charge you accumulate is less likely to be neutralized by the surrounding air, making static shocks more frequent and noticeable when you touch a metal object like a doorknob.

The reason for the shock is the rapid movement of electrons as they try to redistribute themselves to reach a state of electrical neutrality. When you touch a metal object, the excess electrons on your body rapidly transfer to the metal, causing the shock. On a humid day, the air's moisture helps electrons move away from your body more easily, preventing the build-up of a significant static charge.

The Poisson's ratio definition is given as the change in lateral deformation over longitudinal deformation. Mathematically it could be expressed like this,

[tex]\upsilon = \frac{\text{Lateral Strain}}{\text{Longitudinal Strain}}[/tex]

[tex]\upsilon = \frac{(\delta a/a)}{(\delta l/l)}[/tex]

Replacing with our values we would have to,

[tex]\upsilon = \frac{(2.89933-2.9/2.9)}{(9.70710-9.7/97)}[/tex]

[tex]\upsilon = 0.1977[/tex]

Therefore Poisson's ratio is 0.1977.

Answer:

27.1806500378 V

Explanation:

k = Coulomb constant = [tex]8.99\times 10^{9}\ Nm^2/C^2[/tex]

q = Charge = [tex]1.6\times 10^{-19}\ C[/tex]

r = Distance = [tex]5.292\times 10^{-11}\ m[/tex]

Voltage is given by

[tex]V=k\dfrac{q}{r}\\\Rightarrow V=8.99\times 10^9\dfrac{1.6\times 10^{-19}}{5.292\times 10^{-11}}\\\Rightarrow V=27.1806500378\ V[/tex]

The potential difference is 27.1806500378 V

Answer:

a) 2.67 m/s

b) 0.82 J

Explanation:

Amplitude A = 25 cm = 0.25 m

The period of the motion is the inverse of the frequency

[tex]T = \frac{1}{f} = \frac{1}{1.7} = 0.588 s[/tex]

So the angular frequency

[tex]\omega = \frac{2\pi}{T} = \frac{2\pi}{0.588} = 10.68 rad/s[/tex]

The speed at the equilibrium point is the maximum speed, at

[tex]v = \omega A = 10.68 * 0.25 = 2.67 m/s[/tex]

The spring constant can be calculated using the following

[tex]\omega^2 = \frac{k}{m} = \frac{k}{0.23}[/tex]

[tex]k = 0.23\omega^2 = 0.23*10.68^2 = 26.24 N/m[/tex]

The total energy of the oscillation is

[tex]E = kA^2 / 2 = 26.24*0.25^2 / 2 = 0.82 J[/tex]

The sign of the point charge that produces a potential of -2.00 V at a distance of 1.00 mm is negative, and its magnitude is calculated to be approximately -2.22×10-13 C using Coulomb's law.

Explanation:

The question pertains to determining the sign and magnitude of a point charge based on the electric potential it produces at a specific distance. The electric potential (V) at a distance (r) from a point charge (q) is given by the equation V = k * q / r, where k is Coulomb's constant (k = 8.988×109 N·m2/C2). Since the potential is negative (-2.00 V), the point charge must have a negative sign. To find the magnitude, we rearrange the formula to solve for q: q = V * r / k.

Plugging in the values gives q = (-2.00 V * 1.00×10-3 m) / 8.988×109 N·m2/C2, which calculates to a charge magnitude of approximately -2.22×10-13 C.



The probable question is in the image attached.

Answer:

The correct option is 1.  720 m

Explanation:

Projectile Motion

When an object is launched in free air (no friction) with an initial speed vo at an angle [tex]\theta[/tex], it describes a curve which has two components: one in the horizontal direction and the other in the vertical direction. The data provided gives us the initial conditions of the survival package's launch.

[tex]\displaystyle V_o=250\ m/s[/tex]

[tex]\displaystyle \theta =-37^o[/tex]

The initial velocity has these components in the x and y coordinates respectively:

[tex]\displaystyle V_{ox}=250\ cos(-37^o)=199.7\ m/s[/tex]

[tex]\displaystyle V_{oy}=250\ sin(-37^o)=-150.5\ m/s[/tex]

And we know the plane has an altitude of 600 m, so the package will reach ground level when:

[tex]\displaystyle y=-600\ m[/tex]

The vertical distance traveled is given by:

[tex]\displaystyle y=V_{oy}\ t-\frac{g\ t^2}{2}=-600[/tex]

We'll set up an equation to find the time when the package lands

[tex]\displaystyle -150.5t-4.9\ t^2=-600[/tex]

[tex]\displaystyle -4.9\ t^2-150.5\ t+600=0[/tex]

Solving for t, we find only one positive solution:

[tex]\displaystyle t=3.6\ sec[/tex]

The horizontal distance is:

[tex]\displaystyle x=V_{ox}.t=199.7\times3.6=720\ m[/tex]

The correct option is 1.  720 m

The horizontal distance of the point of impact from the plane at the moment of the package's release is approximately 1760 m.

The time taken for the package to reach the ground can be found using the equation y = v0y * t + (1/2) * g * t2, where y is the initial altitude, v0y is the vertical component of the initial velocity, t is the time taken, and g is the acceleration due to gravity. Solving for t gives us a value of approximately 5 seconds. The horizontal distance traveled by the package can be found using the equation x = v0x * t, where x is the horizontal distance, v0x is the horizontal component of the initial velocity, and t is the time taken. Plugging in the values gives us x = 250 m/s * cos(37º) * 5 s, which simplifies to approximately 1760 m. So the horizontal distance of the point of impact is approximately 1760 m.

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The student determine the actual speed of the object when it reaches the ground as 12.52 m/s.

Given data:

The distance from the Earth's surface is, = 10 m.

Time taken to reach the ground is, t = 1.43 s.

The speed of object is, v = 14.3 m/s.

Experimental value of time interval is, t' = 3.2 s.

Use kinematic equation of motion to compute true value for acceleration of the ball as it reaches the ground:

[tex]h=ut+\dfrac{1}{2}a't'^{2} \\\\10=0 \times t+\dfrac{1}{2} \times a' \times 3.2^{2} \\\\a'=\dfrac{20}{3.2^{2}}\\\\a'= 1.95 \;\rm m/s^{2}[/tex]

Now, use the principle of conservation of total energy of system:

Potential energy - work done by air resistance = Kinetic energy

[tex]mgh-(ma) \times h=\dfrac{1}{2}mv^{2} \\\\gh-(a) \times h=\dfrac{1}{2}v^{2} \\\\v=\sqrt{2h(g-a)}[/tex]

Here, v is the actual speed of object while reaching the ground.

Solving as,

[tex]v=\sqrt{2 \times 10(9.8-1.95)}\\\\v=12.52 \;\rm m/s[/tex]

Thus, we can conclude that  the student determine the actual speed of the object when it reaches the ground as 12.52 m/s.

Learn more about the conservation of energy here:

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The actual speed of the object when it reaches the ground is approximately [tex]31.392 \, m/s.[/tex]

The actual speed of the object when it reaches the ground can be determined using the kinematic equation that relates initial velocity, acceleration, and time to the final velocity. Since the object is released from rest, the initial velocity [tex]\( u \)[/tex] is 0 m/s, and the acceleration [tex]\( a \)[/tex] is due to gravity, which is approximately [tex]\( 9.81 \, m/s^2 \)[/tex] near the Earth's surface.

The kinematic equation that relates these quantities to the final velocity [tex]\( v \)[/tex] is:

[tex]\[ v = u + at \][/tex]

Given that [tex]\( u = 0 \, m/s \)[/tex] and [tex]\( a = 9.81 \, m/s^2 \)[/tex], and the time [tex]\( t \)[/tex] to reach the ground is [tex]\( 3.2 \, s \)[/tex], we can substitute these values into the equation to find the actual final velocity:

[tex]\[ v = 0 + (9.81 \, m/s^2)(3.2 \, s) \] \[ v = (9.81)(3.2) \, m/s \] \[ v = 31.392 \, m/s \][/tex]

Therefore, the actual speed of the object when it reaches the ground is approximately [tex]31.392 \, m/s.[/tex]

Answer:

(a) [tex]T_{1}=490N[/tex]

(b) [tex]T_{2}=240N[/tex]

Explanation:

For Part (a)

Given data

The box moves up at steady 5.0 m/s

The mas of box is 50 kg

As ∑Fy=T₁ - mg=0

[tex]T_{1}=mg\\T_{1}=(50kg)(9.8m/s^{2} ) \\T_{1}=490N[/tex]

For Part(b)

Given data

[tex]v_{iy}=5m/s\\ a_{y}=-5.0m/s^{2}[/tex]

As ∑Fy=T₂ - mg=ma

[tex]T_{2}=mg+ma_{y}\\T_{2}=m(g+ a_{y})\\T_{2}=50kg(9.8-5.0) \\T_{2}=240N[/tex]

The tension in the rope varies depending on whether the box is at rest, moving at a constant velocity, or accelerating. The tension equals the weight of the box when it's at rest or moving constantly, but it will be increased or decreased by the net force caused by acceleration when the box is speeding up or slowing down.

If the box is at rest, the tension in the rope is equal to the force of gravity. We can calculate this using the formula T = mg, where m is the mass of the box and g is the acceleration due to gravity. Therefore, T = (50 kg)(9.8 m/s²) = 490 N.

When the box moves upwards with a constant velocity, the tension in the rope also equals the weight of the box (T = mg), so the tension will stay the same at 490 N.

However, when the box is speeding up, the net force is the product of mass and acceleration. In this case, acceleration = 5.0 m/s². Using the equation Fnet = ma, we find that Fnet = (50 kg)(5 m/s²) = 250 N. The total tension now includes both the tension due to the box's weight and the additional force due to the acceleration. Therefore, T = T(g) + Fnet = 490 N + 250 N = 740 N.

Lastly, when the box is slowing down at 5.0 m/s², the net force acts in the opposite direction of the initial velocity. Using the same calculations, we find Fnet = 250 N. But this force now reduces the tension originally caused by the box's weight, so the total tension in the rope becomes T = T(g) - Fnet = 490 N - 250 N = 240 N.

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Final answer:

The problem requires calculating the work done to pump water out of a full swimming pool using given dimensions, the density of water, and gravity.

Explanation:

The question involves finding the amount of work required to pump the water out of a swimming pool when it is full. The dimensions of the pool are given, along with the density of water and the acceleration due to gravity. Using the density of water (1000 kg/m3), the volume of the pool can be calculated to determine the total mass of the water. The work done in pumping the water is found by multiplying the mass by the gravitational constant (9.8 m/s2) and the vertical distance the water needs to be moved (2.2 m, which is the uniform depth of the pool). This distance can be different depending on the location of the pump, but for this problem, we assume the water is being pumped from the very bottom.

Answer:

The ratio of [tex]E_{app}[/tex] and [tex]E_{act}[/tex] is 0.9754

Explanation:

Given that,

Distance z = 4.50 d

First equation is

[tex]E_{act}=\dfrac{qd}{2\pi\epsilon_{0}\times z^3}[/tex]

[tex]E_{act}=\dfrac{Pz}{2\pi\epsilon_{0}\times (z^2-\dfrac{d^2}{4})^2}[/tex]

Second equation is

[tex]E_{app}=\dfrac{P}{2\pi\epsilon_{0}\times z^3}[/tex]

We need to calculate the ratio of [tex]E_{act}[/tex] and [tex]E_{app}[/tex]

Using formula

[tex]\dfrac{E_{app}}{E_{act}}=\dfrac{\dfrac{P}{2\pi\epsilon_{0}\times z^3}}{\dfrac{Pz}{2\pi\epsilon_{0}\times (z^2-\dfrac{d^2}{4})^2}}[/tex]

[tex]\dfrac{E_{app}}{E_{act}}=\dfrac{(z^2-\dfrac{d^2}{4})^2}{z^3(z)}[/tex]

Put the value into the formula

[tex]\dfrac{E_{app}}{E_{act}}=\dfrac{((4.50d)^2-\dfrac{d^2}{4})^2}{(4.50d)^3\times4.50d}[/tex]

[tex]\dfrac{E_{app}}{E_{act}}=0.9754[/tex]

Hence, The ratio of [tex]E_{app}[/tex] and [tex]E_{act}[/tex] is 0.9754

Answer:

Explanation:

Given

mass of first child [tex]m_1=20\ kg[/tex]

mass of second child [tex]m_2=30\ kg[/tex]

Distance between two children is [tex]d=3\ m[/tex]

Suppose light weight child is placed at a distance of x m from Pivot point

therefore

Torque due to heavy child [tex]T_1=m_2g\times (3-x)[/tex]

Torque due to small child [tex]T_2=m_1g\times x[/tex]

Net Torque about Pivot must be zero

Therefore [tex]T_1=T_2[/tex]

[tex]30\times g\times (3-x)=20\times g\times x[/tex]

[tex]9-3x=2x[/tex]

[tex]9=5x[/tex]

[tex]x=\frac{9}{5}[/tex]

[tex]x=1.8\ m[/tex]

Answer:

F_x = 750.7 N

Explanation:

Given:

- Length of the pipe between flanges L = 75 cm

- Weight Flow rate is flow(W) = 230 N/c

- P_1 = 165 KPa

- P_2 = 134 KPa

- P_atm = 101 KPa

- Diameter of pipe D = 0.05 m

Find:

The total force that the flanges must withstand F_x.

Solution:

- Use equation of conservation of momentum.

            (P_1 - P_a)*A + (P_2 - P_a)*A - F_x = flow(m)*( V_2 - V_1)

- From conservation of mass:

                                       A*V_1 = A*V_2

                       V_1 = V_2 ( but opposite in directions)

- Hence,

             (P_1 - P_a)*A + (P_2 - P_a)*A - F_x = - 2*flow(m)*V_1

                                     flow(m) = flow(W) / g

                                     p*A*V_1 = flow(W) / g

                                    V_1 =  flow(W) / g*p*A    

Hence,      

             (P_1 - P_a)*A + (P_2 - P_a)*A - F_x = - 2*flow(W)^2 / g^2*p*A

Hence, compute:

   64*10^3 *pi*0.05^2 /4 + 33*10^3 *pi*0.05^2 /4 - F_x = - 2*(230/9.81)^2 / 997*pi*0.05^2 /4

                            125.6 + 64.7625 - F_x = -560.33

                                        F_x = 750.7 N

The frequency of the alternating current is 394/2π Hz. The resistance of the bulb's filament can be determined using Ohm's Law. The average power delivered to the light bulb can be calculated using the formula P = IV.

(a) The frequency of the alternating current can be calculated using the angular frequency formula ω = 2πf. In this case, the angular frequency is 394 rad/s. So, we can rearrange the formula to find the frequency: f = ω/2π = 394/2π Hz.

(b) The resistance of the bulb's filament can be determined using Ohm's Law, which states that resistance (R) is equal to voltage (V) divided by current (I). In this case, the voltage is 120.0 V and the current is given by I = (0.644 A) sin [(394 rad/s)t].

(c) The average power delivered to the light bulb can be calculated using the formula P = IV, where I is the current and V is the voltage. In this case, the voltage is 120.0 V and the current is given by I = (0.644 A) sin [(394 rad/s)t].

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Answer:

Vab =17.5kV

A = 16.9 cm2

C   = 4.27pF

Explanation:

a) Find the voltage difference:

Vab = Ed

E Electric field

d distance between plates

Vab potential difference

d = 3.5mm

 = 3.5 * 10^(-3) m    

Q = 75.0nC

    = 75 * 10^(-9)  

E = 5.00 * 10^6 V/m

Vab = (5.00 * 10^6) * (3.5 * 10^(-3))

        = 17.5 * 10^3 V

        =17.5kV

b. What is the area of the plate?

 The relation between the electric field and area is given as:

E = Q/(ϵ0 * A)

A = Q/(ϵ0 *E)

Where ϵ0 is the electric constant and equals 8.854 × 10^ (-12) C2/N•m2    

A = 75 * 10^ (-9) / (8.854 × 10^ (-12) (5.00 * 10^6)

 = 1.69 X 10^ (-3) m2

  = 16.9 cm2

c. Find the capacitance

   The equation relating capacitance, area of plate and plate distance is given by:

C = ϵ0 A/d

plug in the values of d, ϵ0 and A above to get the capacitance:

C = (8.854 × 10^ (-12) * 1.69 X 10^ (-3) / 3.5 * 10^ (-3)  

 = 4.27 * 10^ (-12) F

 = 4.27pF

Answer:

3.5 amperes

Explanation:

I = E/R

I = ?

E = 70volts

R = 20 Ohms

Therefore , I = 70/20

= 3.5 amperes

Answer:

Average acceleration = - 2 m/s^2

Explanation:

Given data:

Initial velocity = 11 m/s

Final velocity = 5.0 m/s

duration of change in velocity = 3 sec

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

Average acceleration [tex]= \frac{5 - 11}{3} = -2 m/s^2[/tex]

Average acceleration = - 2 m/s^2

here negative sign indicate that acceleration is proceed in downward direction.