Work Energy and Power: notes and practice questions

An electric ski lift is powered by a motor at the base station. The motor is connected to a 750 V DC power supply by a cable with a total resistance of 0.15 Ω. When operating at full capacity, the motor draws a constant current of 400 A.

(a) Determine the potential difference across the terminals of the motor.

(b) The motor has an efficiency of 92%. Calculate the useful mechanical power output of the motor.

The ski lift carries skiers up a slope of length 1800 m that rises by a vertical height of 500 m. There are 50 chairs on the ascending side. Each empty chair has a mass of 25 kg and carries, on average, 1.5 skiers of average mass 75 kg. A constant resistive force of 12 kN opposes the motion.

(c) Determine the total upward force the motor must provide via the cable to maintain a constant speed.

(d) Estimate the maximum speed, $v$, of the ski lift.

(e) The lift operates continuously. Estimate the maximum number of skiers that can be transported to the top station in one hour.

(f) In an emergency stop, a brake is applied to a large solid steel disc, bringing the lift to a halt from its maximum speed. The total mass of the moving system (chairs, skiers, and cable) is 18 000 kg. Assume all the kinetic energy of the system is converted into thermal energy in the brake disc.

Calculate the temperature rise of the disc.

Data for this question:
Brake disc radius = 0.75 m
Brake disc thickness = 0.10 m
Density of steel = 7850 kg m⁻³
Specific heat capacity of steel = 450 J kg⁻¹ K⁻¹

(g) The speed of a chair is monitored using a radar device at the base station that emits microwaves of frequency 30 GHz. It measures the waves reflected from a chair as it moves away. The frequency shift, $Δf$, for a source moving directly away from a stationary observer can be approximated by the formula $Δf/f ≈ v/c$. In this radar measurement, this formula gives a good approximation for the shift detected.

Calculate the expected frequency shift.

A student is in a perfectly sealed and insulated room. They have two portable electrical devices: a space heater and an air conditioning unit. The air conditioning unit is a heat pump designed to cool the room by transferring thermal energy to the outside via an exhaust hose.

(a) The student switches on the space heater. State and explain the effect on the average temperature of the room.

(b) The student then switches off the heater and switches on the air conditioning unit. However, they leave the exhaust hose inside the sealed room. Explain, by applying the laws of thermodynamics, why this arrangement will not cool the room and will in fact increase its average temperature.

In a specialized medical imaging technique, high-energy X-rays are directed at a target to probe its atomic structure. During this process, a photon with an initial wavelength of $4.00 \times 10^{-12}$ m collides with a stationary electron in the target material. After the collision, the photon's wavelength is observed to have increased by exactly $1.21 \times 10^{-12}$ m.

(a) Calculate the wavelength of the photon after the collision.

(b) Deduce the angle through which the photon has been deflected in this collision.

(c) Explain whether the angle between the original direction of the photon and the final direction of the electron is greater, smaller or equal to your answer in (b).

(d) Determine the kinetic energy of the electron after the collision. Express your answer in keV.

A prototype nuclear reactor uses Plutonium-239 ($_{94}^{239}\text{Pu}$) as fuel. When a neutron is captured by a Plutonium-239 nucleus, one of two possible fission reactions can occur.

Reaction 1: $\text{n} + _{94}^{239}\text{Pu} \rightarrow _{54}^{134}\text{Xe} + _{40}^{103}\text{Zr} + 3\text{n}$

Reaction 2: $\text{n} + _{94}^{239}\text{Pu} \rightarrow _{58}^{144}\text{Ce} + _{36}^{94}\text{Kr} + 2\text{n}$

The following data are available:

(a) Show that the energy released in Reaction 1 is approximately 189 MeV.

(b) Determine the energy released in Reaction 2.

(c) Outline why the neutrons produced in a fission reaction must be slowed down in a thermal nuclear reactor.

The reactor has a useful electrical power output of 550 MW and an overall efficiency of 32%. Assume that all fission events in the reactor follow Reaction 1.

(i) Calculate the total thermal power produced by the fission reactions.

(ii) Calculate the number of Plutonium-239 nuclei that undergo fission every second.

(iii) Calculate the mass of Plutonium-239 that undergoes fission in one week.

(a) A weather balloon is filled with hydrogen gas at a pressure of 1.0 x 10$^5$ Pa and a temperature of 290 K. The volume of the balloon is 5.0 m$^3$. Assume that this sample of hydrogen gas behaves as an ideal gas.

The molar mass of hydrogen (H$_2$) is 2.016 g mol$^{-1}$. Show that the mass of a hydrogen molecule is approximately 3.35 x 10$^{-27}$ kg.

(b) Estimate the average speed of the hydrogen molecules in the balloon.

(c) Calculate the number of hydrogen molecules in the balloon.

(d.i) A hydrogen molecule has an approximate volume of 2.0 x 10$^{-30}$ m$^3$. Calculate the ratio $\frac{\text{total volume of hydrogen molecules}}{\text{volume of hydrogen gas}}$.

(d.ii) Explain, using your answer to (d)(i) and with reference to the kinetic model, why this sample of hydrogen can be assumed to be an ideal gas.

(a.i) A spring has a natural length of $25.0 \text{ cm}$ and negligible mass. Its spring constant is $3.00 \text{ N cm}^{-1}$. The spring obeys Hooke's law when stretched and when compressed.

The spring is horizontal with its left end attached to a rigid support. A $1.50 \text{ kg}$ block is attached to the free end of the spring and can move on a friction-free horizontal surface. The block is pulled $5.0 \text{ cm}$ to the right and held in position.

Calculate the force on the block from the spring when held in position.

(a.ii) Calculate the elastic potential energy stored in the spring when held in position.

(a.iii) The block is released. Calculate the speed of the block when it returns to its original position.

(b.i) The spring is turned to be vertical with the top end attached to a rigid support. The $1.50 \text{ kg}$ block is attached to the bottom of the spring causing it to extend. The block is at rest in its equilibrium position.

Calculate the extension of the spring.

(b.ii) Calculate the elastic potential energy stored in the spring.

(c.i) The block is now pulled down a further $5.0 \text{ cm}$ from the equilibrium position and held in position.

State the total extension of the spring in this new position.

(c.ii) Calculate the total elastic potential energy stored in the spring.

(c.iii) The block is released and accelerates upwards, reaching the equilibrium position with a vertical speed.

Calculate the increase in gravitational potential energy gained by the block as it moves to the equilibrium position.

(c.iv) Calculate the increase in kinetic energy gained by the block as it moves to the equilibrium position.

(c.v) Calculate the speed of the block when it reaches the equilibrium position.

(d) Discuss your answers to (a.iii) and (c.v).

A figure skater is spinning on a horizontal ice surface. There is a small frictional torque exerted by the ice on the skates.

(a) State and explain the effect of this frictional torque on the skater's angular momentum.

(b) While spinning, the skater pulls their arms in towards their body. Ignoring the effects of friction, explain why the total angular momentum of the skater is conserved during this action.

(c) Deduce the effect of pulling their arms in on the skater's angular velocity.

(d) Explain what happens to the skater's rotational kinetic energy when they pull their arms in.

A potter's wheel, designed with most of its mass concentrated at the rim, rotates around a central axis. Its angular speed is increased uniformly from $20 \text{ rad s}^{-1}$ to $80 \text{ rad s}^{-1}$ over a period of $4.0 \text{ s}$. The wheel has a mass of $8.0 \text{ kg}$ and a radius of $30 \text{ cm}$.

Calculate the work done by the motor's torque acting on the wheel.

Calculate the average power supplied to the wheel by the motor during this time.

A component in a high-precision mechanical watch oscillates with simple harmonic motion. The component has a mass $m$. Its displacement $x$ from the equilibrium position at time $t$ is described by the equation $x = A \cos(\omega t)$, where $A$ is the amplitude of oscillation and $\omega$ is the angular frequency.

(a) Determine an expression for the velocity, $v$, of the component as a function of time, $t$.

(b) Hence, show that the kinetic energy of the component is given by $E_K = \frac{1}{2} m \omega^2 A^2 \sin^2(\omega t)$.

(c) The restoring force on the component is provided by a spring of spring constant $k$. Determine an expression for the potential energy, $E_P$, stored in the spring as a function of time, $t$.

(d) Using your answers to (b) and (c), deduce that the total mechanical energy of the oscillator is constant and independent of time.

A rubber bouncy ball with a mass of $75$ g is released from rest at a height of $4.0$ m above a rigid concrete surface. After its first bounce, it reaches a maximum height of $2.8$ m.

(a) Calculate:

i. the speed of the bouncy ball just before it hits the concrete surface.

(a) Calculate:

ii. the speed of the bouncy ball just after it leaves the concrete surface.

(a) Calculate:

iii. the energy lost by the ball in the collision with the concrete surface.

(b) Explain how the law of the conservation of energy applies to this situation if energy has been lost from the ball.