2 hours60 marks

Level 1 - Physics Paper 1

Mechanics, waves and optics, oscillations, collisions, conservation, and fields.

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.

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)

4 marks

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

(b)

3 marks

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

(c)

4 marks

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

(d)

4 marks

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.

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)

4 marks

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

(b)

3 marks

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

(c)

4 marks

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

(d)

3 marks

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

(e)

1 mark

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

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)

4 marks

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

(b)

3 marks

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

(c)

3 marks

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

(d)

3 marks

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

(e)

2 marks

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

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)

3 marks

Find the platform's equilibrium draft.

(b)

3 marks

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

(c)

3 marks

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

(d)

4 marks

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

(e)

2 marks

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.