Question 2
2
A long strip of springy steel is clamped at one end so that the strip is vertical. A mass of 65 g is attached to the free end of the strip, as shown in Fig. 2.1. The mass is pulled to one side and then released. The variation with time t of the horizontal displacement of the mass is shown in Fig. 2.2. The mass undergoes damped simple harmonic motion.
structured6 marks
Question 2(a)
2(a)
4 marks
Question 2(a)(i)
2(a)(i)
Explain what is meant by damping.
Easystructured2 marks
Answer
reduction in energy (of the oscillations) reduction in amplitude / energy of oscillations due to force (always) opposing motion / resistive forces any two of the above, max 2
Question 2(a)(ii)
2(a)(ii)
Suggest, with a reason, whether the damping is light, critical or heavy.
Mediumstructured2 marks
Answer
amplitude is decreasing (very) gradually / oscillations would continue (for a long time) /many oscillations light damping
Question 2(c)
2(c)
After eight complete oscillations of the mass, the amplitude of vibration is reduced from 1.5 cm to 1.1 cm . State and explain whether, after a further eight complete oscillations, the amplitude will be 0.7 cm .
Mediumstructured2 marks
Answer
amplitude reduces exponentially / does not decrease linearly so will be not be 0.7 cm
Question 2
2
A small frictionless trolley is attached to a fixed point A by means of a spring. A second spring is used to attach the trolley to a variable frequency oscillator, as shown in Fig. 2.1. Both springs remain extended within the limit of proportionality. Initially, the oscillator is switched off. The trolley is displaced horizontally along the line joining the two springs and is then released. The variation with time t of the velocity v of the trolley is shown in Fig. 2.2.
structured2 marks
Question 2(b)
2(b)
The oscillator is now switched on. The amplitude of vibration of the oscillator is constant. The frequency f of vibration of the oscillator is varied. The trolley is forced to oscillate by means of vibrations of the oscillator. The variation with f of the amplitude \(a_{0}\) of the oscillations of the trolley is shown in Fig. 2.3. By reference to your answer in (a), state the approximate frequency at which the amplitude is maximum.
Mediumstructured0 marks
Answer
frequency \(=2.4-2.5 \mathrm{~Hz}\)
Question 2(c)
2(c)
The amplitude of the oscillations in (b) may be reduced without changing significantly the frequency at which the amplitude is a maximum. State how this may be done and give a reason for your answer. You may draw on Fig. 2.1 if you wish.
Mediumstructured2 marks
Answer
e.g. attach sheet of card to trolley M1 increases damping / frictional force A1 e.g. reduce oscillator amplitude reduces power/energy input to system
Question 3
3
A microwave cooker uses electromagnetic waves of frequency 2450 MHz . The microwaves warm the food in the cooker by causing water molecules in the food to oscillate with a large amplitude at the frequency of the microwaves.
structured1 marks
Question 3(a)
3(a)
State the name given to this phenomenon.
Easystructured1 marks
Answer
resonance
Question 3
3
A bar magnet of mass 250 g is suspended from the free end of a spring, as illustrated in Fig. 3.1. The magnet hangs so that one pole is near the centre of a coil of wire. The coil is connected in series with a resistor and a switch. The switch is open. The magnet is displaced vertically and then allowed to oscillate. At time t=0, the magnet is oscillating freely. At time \(t=6.0 \mathrm{~s}\), the switch in the circuit is closed. The variation with time t of the vertical displacement y of the magnet is shown in Fig. 3.2.
structured1 marks
Question 3(b)
3(b)
When the switch is closed, the oscillations are damped. Explain, with reference to Fig. 3.2, whether this damping is light, critical or heavy.
Mediumstructured1 marks
Answer
gradual decrease in amplitude, so light (damping) B1
Question 3
3
A bar magnet is suspended from the free end of a helical spring, as illustrated in Fig. 3.1. One pole of the magnet is situated in a coil of wire. The coil is connected in series with a switch and a resistor. The switch is open. The magnet is displaced vertically and then released. As the magnet passes through its rest position, a timer is started. The variation with time t of the vertical displacement y of the magnet from its rest position is shown in Fig. 3.2. At time \(t=4.0 \mathrm{~s}\), the switch is closed.
structured2 marks
Question 3(a)
3(a)
Use Fig. 3.2 to
structured2 marks
Question 3(a)(ii)
3(a)(ii)
state, with a reason, whether the damping after time \(t=4.0 \mathrm{~s}\) is light, critical or heavy,
Mediumstructured2 marks
Answer
amplitude decreases gradually light damping M1 A1 [2]
Question 3
3
An object is suspended from a spring that is attached to a fixed point as shown in Fig. 3.1. The object oscillates vertically with simple harmonic motion about its equilibrium position.
structured2 marks
Question 3(c)
3(c)
The oscillations of the object are now lightly damped.
structured2 marks
Question 3(c)(i)
3(c)(i)
State what is meant by damping.
Easystructured2 marks
Answer
loss of (total) energy (of system) B1 due to resistive forces B1
Question 3
3
A small wooden block (cuboid) of mass m floats in water, as shown in Fig. 3.1. The top face of the block is horizontal and has area A. The density of the water is \(\rho\).
structured1 marks
Question 3(d)
3(d)
The block is now placed in a liquid with a greater density. The block is displaced and released so that it oscillates vertically. The variation with displacement x of the acceleration a of the block is measured for the first half oscillation, as shown in Fig. 3.3.
structured1 marks
Question 3(d)(i)
3(d)(i)
Explain why the maximum negative displacement of the block is not equal to its maximum positive displacement.
Mediumstructured1 marks
Answer
damping due to viscous forces B1
Question 4
4
3 marks
Question 4(c)
4(c)
The variation with time t of the displacement x of the ball in (b) is shown in Fig. 4.2. Some moisture now forms on the track, causing the ball to come to rest after approximately 15 oscillations. On the axes of Fig. 4.2, sketch the variation with time t of the displacement x of the ball for the first two periods after the moisture has formed. Assume the moisture forms at time t=0.
Mediumstructured3 marks
Answer
sketch: time period constant (or increases very slightly) M1 drawn line always 'inside' given loops A1 successive decrease in peak height A1 [3]
Question 3
3
An object is suspended from a vertical spring as shown in Fig. 3.1. The object is displaced vertically and then released so that it oscillates, undergoing simple harmonic motion. Fig. 3.2 The kinetic energy, the potential energy and the total energy of the oscillations are each represented by one of the lines P, Q and R .
structured2 marks
Question 3(c)
3(c)
2 marks
Question 3(c)(i)
3(c)(i)
State the cause of damping.
Easystructured1 marks
Answer
resistive forces B1
Question 3(c)(ii)
3(c)(ii)
A light card is attached to the object. The object is displaced with the same initial amplitude and then released. During each complete oscillation the total energy of the system decreases by 8.0 % of the total energy at the start of that oscillation. Determine the decrease in total energy, in mJ , of the system by the end of the first 6 complete oscillations. energy lost = mJ
Hardstructured0 marks
Answer
\(0.92^{6}\) C1 \[ \begin{aligned} \text { decrease in energy } =6.4-\left(6.4 \times 0.92^{6}\right) =2.5 \mathrm{~mJ} \end{aligned} \] A1
Question 3(c)(iii)
3(c)(iii)
State, with a reason, the type of damping that the card introduces into the system.
Mediumstructured1 marks
Answer
light damping because the amplitude of oscillations gradually reduces or light damping because the system still oscillates B1
Question 4
4
6 marks
Question 4(a)
4(a)
For an oscillating body, state what is meant by
structured2 marks
Question 4(a)(iii)
4(a)(iii)
resonance.
Easystructured2 marks
Answer
maximum amplitude of vibration of oscillating body when forced frequency equals natural frequency (of vibration) B1 B1
Question 4(b)
4(b)
State and explain one situation where resonance is useful.
Mediumstructured2 marks
Answer
e.g. vibration of quartz/piezoelectric crystal (what is vibrating) either for accurate timing (b) or maximise amplitude of ultrasound waves (why it is useful) M1 A1
Question 4(c)
4(c)
In some situations, resonance should be avoided. State one such situation and suggest how the effects of resonance are reduced.
Mediumstructured2 marks
Answer
e.g. vibrating metal panels (what is vibrating) either place strengthening struts across the panel or change shape/area of panel (how it is reduced) M1 A1