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. 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)

4 marks

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

(b)

3 marks

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

(c)

3 marks

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

(d)

3 marks

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

(e)

2 marks

Interpret the signs of the impulse components physically.

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)

3 marks

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

(b)

3 marks

Calculate the tangential acceleration while the cyclist speeds up.

(c)

3 marks

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

(d)

4 marks

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 \[ q=\frac{k\cos\theta-\sin\theta}{\cos\theta+k\sin\theta}. \]

(e)

2 marks

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 \(k=v^2/(rg)\).

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)

3 marks

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

(b)

3 marks

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

(c)

3 marks

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 \[ d\cos\theta\,\Delta\theta=m\Delta\lambda. \]

(d)

3 marks

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.

(e)

3 marks

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 \[ \left(\frac{\Delta\lambda}{\lambda}\right)^2=\left(\frac{\Delta d}{d}\right)^2+(\cot\theta\,\Delta\theta)^2. \]

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)

4 marks

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

(b)

3 marks

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 \[ \omega_{\mathrm{res}}\approx\sqrt{\omega_0^2-2\beta^2}. \]

(c)

3 marks

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.

(d)

2 marks

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

(e)

3 marks

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