Fera Academy

Paper 1

Time2 hours

Marks60

SetSet 1

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

\( g=9.81\,\mathrm{m\,s^{-2}} \)

Section A: Mechanics

1. Package drop, stopping work, and impulse

[15 marks]

A delivery drone travels horizontally at \(11.0\,\mathrm{m\,s^{-1}}\) when it releases a \(0.750\,\mathrm{kg}\) instrument package. The release point is \(28.0\,\mathrm{m}\) above level ground. Air resistance is negligible until the package reaches a foam capture pad.

a) Find the time taken to reach the capture pad and the horizontal distance from the release point to the landing point.

[4 marks]

b) Find the package velocity components immediately before it reaches the pad. Hence find its speed and direction.

[4 marks]

c) The foam brings the package to rest over a distance of \(0.320\,\mathrm{m}\), measured approximately along the incoming path. Estimate the average resisting force exerted by the foam, ignoring the weight impulse during this short stopping interval.

[4 marks]

d) Estimate the impulse exerted by the capture pad on the package. Give the vector impulse using \(+\hat{\imath}\) horizontally forward and \(+\hat{\jmath}\) upward.

[3 marks]

2. Torque, rotational work, and stopping

[15 marks]

A calibration flywheel has moment of inertia \(0.620\,\mathrm{kg\,m^2}\) about its axle. A light cord is wrapped around a drum of radius \(0.180\,\mathrm{m}\) fixed to the flywheel. A technician pulls the free end of the cord with a constant force of \(24.0\,\mathrm{N}\). A bearing friction torque of magnitude \(1.10\,\mathrm{N\,m}\) opposes the rotation.

a) Find the angular acceleration of the flywheel while the cord is being pulled, and find the linear acceleration of the cord.

[5 marks]

b) Starting from rest, the flywheel is pulled until \(2.40\,\mathrm{m}\) of cord has unwound. Use rotational work and energy to find the angular speed at that instant.

[4 marks]

c) After the \(2.40\,\mathrm{m}\) of cord has unwound, the cord leaves the drum and the flywheel slows under the same bearing friction torque. Find the time taken to stop and the additional angle turned.

[4 marks]

d) Explain why increasing the drum radius would increase the driving torque but would not increase the work done by the pull for the same pulled cord length.

[2 marks]

Section B: Waves and Optics

3. Standing waves and grating calibration

[15 marks]

A laboratory rig uses a stretched string and a diffraction grating in the same alignment test. The string has length \(0.640\,\mathrm{m}\), is fixed at both ends, and has linear density \(2.40\times10^{-3}\,\mathrm{kg\,m^{-1}}\). Its fundamental resonance is \(220\,\mathrm{Hz}\). A laser of wavelength \(532\,\mathrm{nm}\) is incident normally on a grating with \(500\,\mathrm{lines\,mm^{-1}}\). A screen is \(1.80\,\mathrm{m}\) from the grating.

a) Derive the relation between the fundamental frequency and the wave speed for this string, then calculate the wave speed.

[3 marks]

b) Calculate the tension in the string.

[3 marks]

c) Find the third-harmonic frequency and the node positions along the string.

[3 marks]

d) Find the grating spacing, the first-order diffraction angle, and the distance on the screen from the central maximum to the first-order maximum.

[4 marks]

e) Find the highest possible diffraction order and explain why the next order is impossible.

[2 marks]

Section C: Oscillations and Collisions, Conservation and Fields

4. Damped sensor motion and floating equilibrium

[15 marks]

A floating environmental buoy contains a vertical spring-mass accelerometer. The oscillating sensor mass is \(0.450\,\mathrm{kg}\), the spring constant is \(18.0\,\mathrm{N\,m^{-1}}\), and the linear damping coefficient is \(0.180\,\mathrm{kg\,s^{-1}}\). After a wave passes, the sensor is displaced by \(0.0600\,\mathrm{m}\) from equilibrium and released from rest. The complete buoy, including the sensor, has mass \(3.65\,\mathrm{kg}\) and a vertical cylindrical hull of cross-sectional area \(3.00\times10^{-2}\,\mathrm{m^2}\). The damping rate is \[ \beta=\frac{b}{2m} \] and the damped angular frequency is \[ \omega_d=\sqrt{\omega_0^2-\beta^2}. \] For underdamped motion, the amplitude envelope is \[ A(t)=A_0e^{-\beta t}. \]

a) Calculate the undamped angular frequency and the damping rate. Decide whether the motion is underdamped.

[4 marks]

b) Find the damped period and write the amplitude envelope for the motion.

[3 marks]

c) Estimate the amplitude after three damped periods and the mechanical energy lost over those three periods.

[4 marks]

d) In fresh water of density \(1000\,\mathrm{kg\,m^{-3}}\), find the depth of the cylindrical hull below the waterline.

[3 marks]

e) State how the immersion depth changes if the buoy is placed in seawater of density \(1025\,\mathrm{kg\,m^{-3}}\).

[1 mark]

END