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 .
structured2 marks
Question 1(a)
1(a)
For the oscillations of the magnet, use Fig. 1.2 to
structured2 marks
Question 1(a)(i)
1(a)(i)
determine the angular frequency \(\omega\),
Mediumstructured2 marks
Answer
either \(\omega=2 \pi / T\) or \(\omega=2 \pi f\) and \(f=1 / T \quad\) C1
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.
structured1 marks
Question 2(b)
2(b)
1 marks
Question 2(b)(i)
2(b)(i)
Use Fig. 2.2 to determine the frequency of vibration of the mass.
Easystructured1 marks
Answer
frequency =1 / 0.3 allow points taken from time axis giving \(f=3.45 \mathrm{~Hz}\)
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.
structured5 marks
Question 2(a)
2(a)
5 marks
Question 2(a)(i)
2(a)(i)
Using Fig. 2.2, state two different times at which 1. the displacement of the trolley is zero, 2. the acceleration in one direction is maximum.
Easystructured2 marks
Answer
1. \(0.1 \mathrm{~s}, 0.3 \mathrm{~s}, 0.5 \mathrm{~s}\), etc (any two) 2. either \(0,0.4 \mathrm{~s}, 0.8 \mathrm{~s}, 1.2 \mathrm{~s}\) or \(0.2 \mathrm{~s}, 0.6 \mathrm{~s}, 1.0 \mathrm{~s}\) (any two)
Question 2(a)(ii)
2(a)(ii)
Determine the frequency of oscillation of the trolley.
Easystructured2 marks
Answer
period \(=0.4 \mathrm{~s}\) frequency \(=(1 / 0.4=) 2.5 \mathrm{~Hz}\)
Question 2(a)(iii)
2(a)(iii)
The variation with time of the displacement of the trolley is sinusoidal. The variation with time of the velocity of the trolley is also sinusoidal. State the phase difference between the displacement and the velocity.
Easystructured1 marks
Answer
phase difference \(=90^{\circ}\) or \(\frac{1}{2} \pi \mathrm{rad}\)
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(b)
2(b)
At two points in the oscillation of the ball, its kinetic energy is equal to the potential energy stored in the springs. Calculate the magnitude of the displacement at which this occurs. displacement = cm
Mediumstructured3 marks
Answer
\(E_{\mathrm{K}}=E_{\mathrm{P}}\) \(1 / 2 m \omega^{2}\left(a^{2}-x^{2}\right)=1 / 2 m \omega^{2} x^{2}\) or \(E_{\mathrm{K}}\) or \(E_{\mathrm{P}}=3.5 \mathrm{~mJ}\) C1 \(=a / \sqrt{ } 2=2.8 / \sqrt{ } 2\) or \(E_{\mathrm{K}}=1 / 2 m \omega^{2}\left(a^{2}-x^{2}\right)\) or \(E_{\mathrm{P}}=1 / 2 m \omega^{2} x^{2}\) \(=2.0 \mathrm{~cm}\) C1 A1 ( \(E_{\mathrm{K}}\) or \(E_{\mathrm{P}}=7.0 \mathrm{~mJ}\) scores 0/3) Allow:
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
structured2 marks
Question 2(c)(ii)
2(c)(ii)
the kinetic energy of the ball (label this line K ),
Easystructured2 marks
Answer
graph: reasonable curve B1 with maximum at ( 0,7.0 ) end-points of line at ( -2.8,0 ) and ( +2.8,0 ) B1
Question 10
10
A mass on the end of a spring bounces up and down as shown, after being released at time t=0. Which graph shows how the velocity varies with time?
Hardmcq1 marks
Answer
A
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 .
structured6 marks
Question 3(a)
3(a)
The mass is released at time t=0. For the oscillations of the mass,
structured6 marks
Question 3(a)(i)
3(a)(i)
calculate the angular frequency \(\omega\),
Easystructured2 marks
Answer
\(\omega=2 \pi / T\) (allow use of \(f=1.5 \mathrm{~Hz}\) to give \(\omega=9.4 \mathrm{rad} \mathrm{s}^{-1}\) )
Question 3(a)(ii)
3(a)(ii)
determine numerical equations for the variation with time t of 1. the displacement x in cm , 2. the speed v in \(\mathrm{ms}^{-1}\).
Mediumstructured4 marks
Answer
1. \(x=2.1 \cos 9.1 t\) 2.1 and 9.1 numerical values B1 use of cos B1 2. \(\quad v_{0}=2.1 \times 10^{-2} \times 9.1\) (allow ecf of value of \(x_{0}\) from (ii)1.) \(=0.19 \mathrm{~m} \mathrm{~s}^{-1}\) \(v=v_{0} \sin 9.1 t\) (allow \(\cos 9.1 t\) if \(\sin\) used in (ii)1.)
Question 3
3
7 marks
Question 3(a)
3(a)
Define simple harmonic motion.
Easystructured2 marks
Answer
acceleration / force proportional to displacement from a fixed point M1 acceleration / force (always) directed towards that fixed point / in opposite direction to displacement
Question 3(b)
3(b)
A tube, sealed at one end, has a total mass m and a uniform area of cross-section A. The tube floats upright in a liquid of density \(\rho\) with length L submerged, as shown in Fig. 3.1a. The tube is displaced vertically and then released. The tube oscillates vertically in the liquid. At one time, the displacement is x, as shown in Fig. 3.1b. Theory shows that the acceleration a of the tube is given by the expression
structured5 marks
Question 3(b)(i)
3(b)(i)
Explain how it can be deduced from the expression that the tube is moving with simple harmonic motion.
Mediumstructured2 marks
Answer
\(A \rho g / m\) is a constant and so acceleration proportional to \(x \quad B1\) negative sign shows acceleration towards a fixed point / in opposite direction to displacement B1
Question 3(b)(ii)
3(b)(ii)
The tube, of area of cross-section \(4.5 \mathrm{~cm}^{2}\), is floating in water of density \(1.0 \times 10^{3} \mathrm{~kg} \mathrm{~m}^{-3}\). Calculate the mass of the tube that would give rise to oscillations of frequency 1.5 Hz . mass =
Hardstructured3 marks
Answer
\(\omega^{2}=(A \rho g / m) \quad\) C1 \(\omega=2 \pi f\) C1 \((2 \times \pi \times 1.5)^{2}=\left(\left\{4.5 \times 10^{-4} \times 1.0 \times 10^{3} \times 9.81\right\} / m\right) \quad\) C1 \(m=50 \mathrm{~g}\) A1
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.
structured2 marks
Question 3(a)
3(a)
For the oscillating magnet, use data from Fig. 3.2 to determine, to two significant figures:
structured2 marks
Question 3(a)(i)
3(a)(i)
the frequency f
Easystructured2 marks
Answer
e.g. period =6 / 2.5 C1 frequency \(=0.42 \mathrm{~Hz}\) 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.
structured4 marks
Question 3(b)
3(b)
The frequency of the oscillator is fixed, and the amplitude of the oscillations is gradually increased.
structured4 marks
Question 3(b)(i)
3(b)(i)
Calculate the maximum amplitude of the oscillations so the object does not lose contact with the platform. amplitude = m
Mediumstructured2 marks
Answer
use of acceleration \(=9.81 \mathrm{~m} \mathrm{~s}^{-2}\) \[ x_{0}=9.81 / 39^{2} \] C1 \(=6.4 \times 10^{-3} \mathrm{~m}\) A1
Question 3(b)(ii)
3(b)(ii)
The amplitude of the oscillations is increased so it is greater than the value in (b)(i). State and explain the position in an oscillation where the object first loses contact with the platform.
Mediumstructured2 marks
Answer
at top of oscillation B1 any one point from: where the downward acceleration first exceeds free-fall acceleration where the greatest downwards acceleration occurs where the resultant force is the maximum downwards where the contact force is a minimum B1
Question 4
4
3 marks
Question 4(b)
4(b)
A ball of mass 65 g is thrown vertically upwards from ground level with a speed of \(16 \mathrm{~m} \mathrm{~s}^{-1}\). Air resistance is negligible.
structured3 marks
Question 4(b)(ii)
4(b)(ii)
The ball takes time t to reach maximum height. For time \(\frac{t}{2}\) after the ball has been thrown, calculate the ratio
Hardstructured3 marks
Answer
speed at t=1 / 2 total time \(=8\left(\mathrm{~ms}^{-1}\right)\) or total t=1.63 or \(t_{1 / 2}=0.815 \mathrm{~s} \quad\) C1 KE is \(\frac{1}{4}\) or h at \(t_{1 / 2}=9.78(\mathrm{~m}) \quad\) C1 and PE is \(\frac{3}{4}\) of max ratio \(=3 \quad\) or ratio \(=9.78 / 3.26=3 \quad\) A1 [3]