Fera Academy

Paper 1

Time2 hours

Marks60

SetSet 4

PaperLevel 1 - Physics Paper 1

Information

Section A: Mechanics
Section B: Waves and Optics
Section C: Oscillations

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. Projectile launch, stopping work, and impulse

[15 marks]

A robotic arm releases a sealed sample cartridge of mass \(0.180\,\mathrm{kg}\) from the end of a horizontal conveyor. At release the cartridge is \(3.20\,\mathrm{m}\) horizontally from the centre of a padded catch tray, and the tray surface is \(0.080\,\mathrm{m}\) below the release point. The launch speed is \(5.80\,\mathrm{m\,s^{-1}}\) at \(32.0^\circ\) above the horizontal. Air resistance is negligible until the cartridge enters the tray padding, which brings it to rest over \(0.060\,\mathrm{m}\) measured along the direction of its motion.

a) Show that the projectile path carries the cartridge to the catch tray.

[4 marks]

b) Calculate the velocity of the cartridge just before it touches the tray padding.

[3 marks]

c) Estimate the average stopping force exerted by the tray padding. State the main approximation.

[3 marks]

d) Find the impulse vector exerted by the padding on the cartridge, using \(\hat{\imath}\) horizontally toward the tray and \(\hat{\jmath}\) upward.

[3 marks]

e) Interpret the signs of the impulse components physically.

[2 marks]

2. Banked circular motion and changing speed

[15 marks]

A cyclist rides on a banked indoor velodrome bend of radius \(28.0\,\mathrm{m}\). The track is banked at \(12.0^\circ\) to the horizontal. For the tyres on the timber surface, the coefficient of static friction is \(0.350\). During one half-turn, the cyclist speeds up uniformly from \(9.00\,\mathrm{m\,s^{-1}}\) to \(15.0\,\mathrm{m\,s^{-1}}\). Treat the cyclist and bicycle as a point mass.

a) Find the speed for which no cross-slope friction is needed.

[3 marks]

b) Calculate the tangential acceleration while the cyclist speeds up.

[3 marks]

c) Find the radial acceleration and the magnitude of the total acceleration when the speed is \(15.0\,\mathrm{m\,s^{-1}}\).

[3 marks]

d) Determine the direction of the cross-slope friction at high speed and test whether \(15.0\,\mathrm{m\,s^{-1}}\) is possible without slipping. Let \(k=v^2/(rg)\) and \(q=f/N\). For friction down the slope, the required friction ratio is \[ q=\frac{k\cos\theta-\sin\theta}{\cos\theta+k\sin\theta}. \]

[4 marks]

e) Find the safe-speed range for the banked bend. At the high-speed limit, the equation for \(k_{\max}\) is \[ k_{\max}=\frac{\sin\theta+\mu_s\cos\theta}{\cos\theta-\mu_s\sin\theta}, \] where \(k=v^2/(rg)\).

[2 marks]

Section B: Waves and Optics

3. Grating spectrometer resolution and uncertainty

[15 marks]

A diffraction grating spectrometer is used at normal incidence to compare two close green spectral lines of wavelengths \(540.0\,\mathrm{nm}\) and \(541.8\,\mathrm{nm}\). The grating has \(750\,\mathrm{lines\,mm^{-1}}\), and a screen is placed \(1.50\,\mathrm{m}\) from the grating. The illuminated width of the grating is \(4.00\,\mathrm{mm}\).

a) Find the grating spacing and the highest possible order for these wavelengths.

[3 marks]

b) Calculate the first-order and second-order angles for the \(540.0\,\mathrm{nm}\) line.

[3 marks]

c) Estimate the first-order separation of the two spectral lines on the screen. For a small wavelength difference at fixed order, the angular separation is given by \[ d\cos\theta\,\Delta\theta=m\Delta\lambda. \]

[3 marks]

d) Use resolving power to decide whether the two wavelengths can be resolved in first and second order. The resolving power of a grating is \[ R=mN \] and also \[ R=\frac{\lambda}{\Delta\lambda_{\min}}, \] where \(N\) is the number of illuminated grating lines.

[3 marks]

e) If the grating line density is \(750\pm3\,\mathrm{lines\,mm^{-1}}\) and the first-order angle reading has uncertainty \(\pm0.05^\circ\), estimate the uncertainty in a wavelength determined from the first-order angle. The fractional uncertainty is \[ \left(\frac{\Delta\lambda}{\lambda}\right)^2=\left(\frac{\Delta d}{d}\right)^2+(\cot\theta\,\Delta\theta)^2. \]

[3 marks]

Section C: Oscillations

4. Driven damped oscillator in a seismometer

[15 marks]

A laboratory seismometer contains a proof mass of \(0.800\,\mathrm{kg}\) attached to a spring of stiffness \(320\,\mathrm{N\,m^{-1}}\). A magnetic damper provides approximately linear damping, so the relative displacement \(x\) of the mass can be modelled as a driven damped oscillator. In one damping setting, a free oscillation has its amplitude reduced by a factor of 2 after 18 complete cycles. Later, a steady sinusoidal ground motion drives the instrument near resonance and the relative displacement amplitude is \(12.0\,\mathrm{mm}\).

a) Write the driven damped oscillator equation for the seismometer and calculate its undamped natural frequency.

[4 marks]

b) Estimate the damping rate and the resonant angular frequency for this lightly damped setting. The free-oscillation amplitude envelope is \[ A=A_0e^{-\beta t}. \] For light damping, the resonant angular frequency is approximately \[ \omega_{\mathrm{res}}\approx\sqrt{\omega_0^2-2\beta^2}. \]

[3 marks]

c) Estimate the percentage of mechanical energy lost per cycle and the quality factor \(Q\). The logarithmic decrement per cycle is \[ \delta=\beta T. \] For weak damping, the quality factor is approximately \[ Q\approx\frac{\pi}{\delta}. \] The oscillator energy is proportional to amplitude squared.

[3 marks]

d) Find the mechanical energy stored in the oscillator at the stated steady-state amplitude.

[2 marks]

e) Compare this setting with one in which the damping coefficient is doubled. Discuss the effect on \(Q\), energy loss per cycle, and resonant amplitude.

[3 marks]

END