Fera Academy

Paper 1

Time2 hours

Marks60

SetSet 2

PaperLevel 1 - Physics Paper 1

Information

Section A: Mechanics
Section B: Waves and Optics
Section C: Oscillations and Collisions, Conservation and Fields

Candidate name

Candidate number

Instructions

Attempt all questions.
All main questions carry the same number of marks.
Show enough working to make the model, assumptions, and sign conventions clear.
Begin each main question on a new page when working on paper.
Use the constants printed in this paper where relevant.
Answer spaces are provided after each question part.

Constants

Gravitational acceleration	\( g=9.81\,\mathrm{m\,s^{-2}} \)
Speed of light	\( c=3.00\times10^8\,\mathrm{m\,s^{-1}} \)
Elementary charge	\( e=1.60\times10^{-19}\,\mathrm{C} \)
Electron mass	\( m_e=9.11\times10^{-31}\,\mathrm{kg} \)
Planck constant	\( h=6.63\times10^{-34}\,\mathrm{J\,s} \)
Permittivity of free space	\( \epsilon_0=8.85\times10^{-12}\,\mathrm{F\,m^{-1}} \)
Magnetic constant	\( \mu_0=1.26\times10^{-6}\,\mathrm{N\,A^{-2}} \)
Boltzmann constant	\( k_B=1.38\times10^{-23}\,\mathrm{J\,K^{-1}} \)

Section A: Mechanics

1. Rescue sled on a curved valley track

[15 marks]

A rescue sled of total mass \(180\,\mathrm{kg}\) crosses a mountain valley on a smooth curved track. It is released from rest at a station \(14.0\,\mathrm{m}\) above the lowest point of the track. At the lowest point the track is locally circular with radius \(22.0\,\mathrm{m}\). Treat the sled as a particle and neglect rolling resistance.

a) Use energy conservation to find the sled's speed at the lowest point.

[4 marks]

b) Calculate the magnitude and direction of the sled's centripetal acceleration at the lowest point.

[3 marks]

c) Find the normal force exerted by the track on the sled at the lowest point.

[4 marks]

d) The track is rated for a maximum normal force of \(4.50\times10^3\,\mathrm{N}\) on this loaded sled at the lowest point. Find the maximum safe speed there and the corresponding maximum release height above the lowest point.

[4 marks]

2. Winch drum with changing torque

[15 marks]

A motor drives a winch drum of radius \(0.120\,\mathrm{m}\) and moment of inertia \(0.360\,\mathrm{kg\,m^2}\). A light cord wrapped around the drum lifts a \(25.0\,\mathrm{kg}\) crate vertically without slipping. From rest, the motor torque is \(\tau(t)=36.0+4.00t\,\mathrm{N\,m}\), where \(t\) is in seconds. Friction in the bearings is negligible.

a) Derive an expression for the angular acceleration of the drum as a function of time.

[4 marks]

b) Find the speed of the cord and crate after \(3.00\,\mathrm{s}\).

[3 marks]

c) Find the work done by the motor during the first \(3.00\,\mathrm{s}\).

[4 marks]

d) Calculate the instantaneous power delivered by the motor at \(t=3.00\,\mathrm{s}\).

[3 marks]

e) State what would happen if the applied torque were less than \(mgR\).

[1 mark]

Section B: Waves and Optics

3. Camera calibration and diffraction limit

[15 marks]

A compact camera is calibrated by photographing a flat test chart \(2.40\,\mathrm{m}\) in front of a thin converging lens of focal length \(12.0\,\mathrm{mm}\). The entrance pupil is a circular aperture of diameter \(3.00\,\mathrm{mm}\), and the calibration light has wavelength \(550\,\mathrm{nm}\). For a circular aperture, the Airy/Rayleigh angular radius is \[ \theta_R=1.22\frac{\lambda}{D}, \] where \(D\) is the aperture diameter.

a) Find the image distance from the lens to the sensor plane.

[4 marks]

b) Find the magnification and the image height of a \(1.20\,\mathrm{m}\) high calibration chart.

[3 marks]

c) Estimate the diffraction-limited angular radius of the Airy disk and its radius on the sensor.

[3 marks]

d) Using the Rayleigh criterion, decide whether two fine marks \(0.800\,\mathrm{mm}\) apart on the chart can be resolved by the aperture.

[3 marks]

e) State how the diffraction-limited resolution changes if the aperture diameter is reduced to \(1.50\,\mathrm{mm}\).

[2 marks]

Section C: Oscillations and Collisions, Conservation and Fields

4. Floating platform oscillations

[15 marks]

A small floating service platform has vertical sides, horizontal waterline area \(3.20\,\mathrm{m^2}\), and mass \(240\,\mathrm{kg}\). It floats in fresh water of density \(1000\,\mathrm{kg\,m^{-3}}\). For small vertical displacements, neglect damping and assume the waterline area remains constant.

a) Find the platform's equilibrium draft.

[3 marks]

b) Show that a small vertical displacement gives simple harmonic motion and find the effective spring constant.

[3 marks]

c) Calculate the platform's angular frequency and frequency of vertical oscillation.

[3 marks]

d) A \(160\,\mathrm{kg}\) payload is placed on the platform. Find the new equilibrium draft and the new oscillation frequency.

[4 marks]

e) With the payload on board, the platform is pushed \(0.0400\,\mathrm{m}\) below its new equilibrium position and released from rest. Find the oscillation energy and maximum speed.

[2 marks]

END