Question 1
1
A light spring is suspended from a fixed point. A bar magnet is attached to the end of the spring, as shown in Fig. 1.1. In order to shield the magnet from draughts, a cardboard cup is placed around the magnet but does not touch it. The magnet is displaced vertically and then released. The variation with time t of the vertical displacement y of the magnet is shown in Fig. 1.2. The mass of the magnet is 130 g .
structured4 marks
Question 1(a)
1(a)
For the oscillations of the magnet, use Fig. 1.2 to
structured2 marks
Question 1(a)(ii)
1(a)(ii)
show that the maximum kinetic energy of the oscillating magnet is 6.4 mJ .
Mediumstructured2 marks
Answer
kinetic energy \(=1 / 2 m \omega^{2} x_{0}{ }^{2}\) or \(v=\omega x_{0}\) and \(\frac{1}{2} m v^{2} \quad\) C1
Question 1(b)
1(b)
The cardboard cup is now replaced with a cup made of aluminium foil. During 10 complete oscillations of the magnet, the amplitude of vibration is seen to decrease to 0.75 cm from that shown in Fig. 1.2. The change in angular frequency is negligible.
structured2 marks
Question 1(b)(ii)
1(b)(ii)
Show that the loss in energy of the oscillating magnet is 4.8 mJ .
Mediumstructured2 marks
Answer
either use of \(\frac{1}{2} m \omega^{2} x_{0}{ }^{2}\) and \(x_{0}=0.75 \mathrm{~cm}\) or \(x_{0}\) is halved so \(\frac{1}{4}\) energy to give new energy \(=1.6 \mathrm{~mJ}\) either loss in energy =6.4-1.6 or loss \(=\frac{3}{4} \times 6.4\) giving loss \(=4.8 \mathrm{~mJ}\)
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.
structured2 marks
Question 2(b)
2(b)
2 marks
Question 2(b)(ii)
2(b)(ii)
Hence show that the initial energy stored in the steel strip before the mass is released is approximately 3.2 mJ .
Mediumstructured2 marks
Answer
energy \(=1 / 2 m v^{2}\) and \(v=\omega a\) \(=1 / 2 \times 0.065 \times(2 \pi / 0.3)^{2} \times\left(1.5 \times 10^{-2}\right)^{2}\) \(=3.2 \mathrm{~mJ}\)
Question 2
2
A ball of mass 37 g is held between two fixed points A and B by two stretched helical springs, as shown in Fig. 2.1. The ball oscillates along the line AB with simple harmonic motion of frequency 3.5 Hz and amplitude 2.8 cm .
structured5 marks
Question 2(a)
2(a)
Show that the total energy of the oscillations is 7.0 mJ .
Mediumstructured2 marks
Answer
(allow \(2 \pi \times 3.5\) shown as \(7 \pi\) ) Energy \(=1 / 2 m v^{2}\) and \(v=r \omega\) Correct substitution Energy \(=7.0 \times 10^{-3} \mathrm{~J}\)
Question 2(c)
2(c)
On the axes of Fig. 2.2 and using your answers in (a) and (b), sketch a graph to show the variation with displacement x of
structured3 marks
Question 2(c)(i)
2(c)(i)
the total energy of the system (label this line T ),
Easystructured1 marks
Answer
graph: horizontal line, y-intercept \(=7.0 \mathrm{~mJ}\) with end-points of line at +2.8 cm and -2.8 cm B1
Question 2(c)(iii)
2(c)(iii)
the potential energy stored in the springs (label this line P ).
Easystructured2 marks
Answer
graph: inverted version of (ii) M1 with intersections at (-2.0, 3.5) and (+2.0, 3.5) A1 (Allow marks in (iii), but not in (ii), if graphs K \& P are not labelled)
Question 3
3
A mass of 78 g is suspended from a fixed point by means of a spring, as illustrated in Fig. 3.1. The stationary mass is pulled vertically downwards through a distance of 2.1 cm and then released. The mass is observed to perform simple harmonic motion with a period of 0.69 s .
structured2 marks
Question 3(b)
3(b)
Calculate the total energy of oscillation of the mass. energy = J
Mediumstructured2 marks
Answer
energy = either \(1 / 2 m v_{0}{ }^{2}\) or \(1 / 2 m \omega^{2} x_{0}{ }^{2}\)
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.
structured3 marks
Question 3(a)
3(a)
For the oscillating magnet, use data from Fig. 3.2 to determine, to two significant figures:
structured3 marks
Question 3(a)(ii)
3(a)(ii)
the energy of the oscillations during the time interval t=0 to \(t=6.0 \mathrm{~s}\).
Mediumstructured3 marks
Answer
energy \(=1 / 2 m \omega^{2} y_{0}{ }^{2}\) C1 \(=\frac{1}{2} \times 0.25 \times 4 \pi^{2} \times 0.42^{2} \times\left(1.5 \times 10^{-2}\right)^{2}\) \[ \text { energy }=1 / 2 m \times 4 \pi^{2} f^{2} y_{0}^{2} \] C1 energy \(=1.9 \times 10^{-4} \mathrm{~J}\) A1
Question 3
3
A small object of mass 24 g rests on a platform. The platform is attached to an oscillator, as shown in Fig. 3.1. The oscillator moves the platform up and down.
structured3 marks
Question 3(a)
3(a)
The total energy of the oscillations of the object is \(2.2 \times 10^{-4} \mathrm{~J}\). In one oscillation the object travels a total distance of 14 mm . Calculate the angular frequency \(\omega\) of the oscillations.
Mediumstructured3 marks
Answer
\(E=1 / 2 m \omega^{2} x_{\mathrm{o}^{2}}\) C1 \(2.2 \times 10^{-4}=1 / 2 \times 24 \times 10^{-3} \times\left(14 \times 10^{-3} / 4\right)^{2} \times \omega^{2}\) C1 \(\omega=39 \mathrm{rad} \mathrm{s}^{-1}\) A1
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 \(\rho\).
structured3 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.
structured3 marks
Question 3(d)(ii)
3(d)(ii)
The mass of the block is 0.57 kg . Use Fig. 3.3 to determine the decrease \(\Delta E\) in energy of the oscillation for the first half oscillation. E= J
Hardstructured3 marks
Answer
\((E=) 1 / 2 m \omega^{2} x_{0}{ }^{2}\) C1 \(\omega^{2}=(-)\) gradient C1 \[ \begin{aligned} (E=) 1 / 2 m \omega^{2}\left(x_{1}^{2}-x_{2}^{2}\right) =1 / 2 \times 0.57 \times(2.3 / 0.020)\left(0.020^{2}-0.016^{2}\right) =4.7 \times 10^{-3} \mathrm{~J} \end{aligned} \] A1
Question 3
3
A metal ball is suspended from a fixed point by means of a string, as illustrated in Fig. 3.1. The ball is given a small displacement and then released. The variation with time t of the displacement x of the ball is shown in Fig. 3.2.
structured4 marks
Question 3(b)
3(b)
The variation with displacement x of the potential energy \(E_{\mathrm{P}}\) of the oscillations of the ball is shown in Fig. 3.3.
structured4 marks
Question 3(b)(i)
3(b)(i)
On the axes of Fig. 3.3, sketch a graph to show the variation with displacement x of the kinetic energy of the ball.
Mediumstructured2 marks
Answer
sketch: curve from \(( \pm 1.5,0)\) passing through (0,25) M1 reasonable shape (curved with both intersections between \(y=12.0 \rightarrow 13.0\) )
Question 3(b)(ii)
3(b)(ii)
The amplitude of the oscillations reduces over a long period of time. After many oscillations, the amplitude of the oscillations is 0.60 cm . Use Fig. 3.3 to determine the total energy of the oscillations of the ball for oscillations of amplitude 0.60 cm . Explain your working. energy = J
Hardstructured2 marks
Answer
at max. amplitude potential energy is total energy B1 total energy \(=4.0 \mathrm{~mJ}\) B1
Question 3
3
A ball is held between two fixed points A and B by means of two stretched springs, as shown in Fig. 3.1. The ball is free to oscillate along the straight line AB . The springs remain stretched and the motion of the ball is simple harmonic. The variation with time t of the displacement x of the ball from its equilibrium position is shown in Fig. 3.2.
structured3 marks
Question 3(c)
3(c)
Calculate the displacement of the ball at which its kinetic energy is equal to one half of the maximum kinetic energy. cm
Hardstructured3 marks
Answer
either kinetic energy \(=1 / 2 m \omega^{2}\left(x_{0}{ }^{2}-x^{2}\right)\) or potential energy \(=1 / 2 m \omega^{2} x^{2}\) and potential energy = kinetic energy \(1 / 2 m \omega^{2}\left(x_{0}-x^{2}\right)=1 / 2 \times 1 / 2 m \omega^{2} x_{0}{ }^{2}\) or \(1 / 2 m \omega^{2} x^{2}=1 / 2 \times 1 / 2 m \omega^{2} x_{0}{ }^{2}\) \(x_{0}{ }^{2}=2 x^{2}\) \(x=x_{0} / \sqrt{ } 2=1.7 / \sqrt{ } 2\) \(=1.2 \mathrm{~cm}\)
Question 4
4
A small metal ball is suspended from a fixed point by means of a string, as shown in Fig. 4.1. The ball is pulled a small distance to one side and then released. The variation with time t of the horizontal displacement x of the ball is shown in Fig. 4.2. The motion of the ball is simple harmonic.
structured3 marks
Question 4(b)
4(b)
The maximum kinetic energy of the ball is \(E_{\mathrm{K}}\). On the axes of Fig. 4.3, sketch a graph to show the variation with time t of the kinetic energy of the ball for the first 1.0 s of its motion.
Mediumstructured3 marks
Answer
sinusoidal wave with all values positive B1 all values positive, all peaks at \(E_{\mathrm{K}}\) and energy =0 at \(t=0 \quad\) B1 period \(=0.30 \mathrm{~s}\) B1