Simple harmonic systemsAQA A-Level Physics: Subtopic test

1. 1
   A 0.250 kg mass hangs from a light spring of spring constant 40 N m⁻¹. The mass is pulled down a small distance and released, so that it oscillates vertically with simple harmonic motion.

   (a)

   Calculate the time period of the oscillations.

   [1 mark]

   • A0.25 s
   • B0.50 s
   • C2.0 s
   • D0.080 s

   (b)

   The 0.250 kg mass is replaced by a 1.00 kg mass and the oscillation is repeated with a small amplitude. How does the time period change?

   [1 mark]

   • AIt halves
   • BIt is unchanged
   • CIt doubles
   • DIt quadruples

   (c)

   The apparatus is taken to the Moon, where the gravitational field strength is smaller, and the oscillation is repeated with the same mass and spring. State and explain the effect on the time period.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A student sets up a simple pendulum in a laboratory using a small dense bob on a light inextensible string. The distance from the point of suspension to the centre of the bob is 0.640 m. The local gravitational field strength is 9.81 N kg⁻¹.

   (a)

   The pendulum swings through a small angle. What is the time period?

   [1 mark]

   • A0.26 s
   • B0.80 s
   • C3.2 s
   • D1.60 s

   (b)

   The student says that the motion can be treated as simple harmonic provided a condition is met. Which condition is correct?

   [1 mark]

   • AThe angular amplitude is small, so that $\sin\theta \approx \theta$ with $\theta$ in radians
   • BThe mass of the bob is large compared with the mass of the string
   • CThe angular amplitude is large, so that the restoring force is constant
   • DThe string is long, so that the bob moves in a straight line

   (c)

   A pendulum clock keeps correct time at sea level. It is taken to the top of a high mountain, where $g$ is slightly smaller, and its length is not changed. Explain why the clock runs slow.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A uniform U-tube of cross-sectional area $A$ contains liquid of density $\rho$ with a total column length $L$ = 0.360 m. When the liquid in one arm is pushed down by a displacement $x$, the liquid in the other arm rises by $x$, and the unbalanced weight of liquid acting to restore equilibrium is $2\rho A g x$. Viscous effects are negligible.

   (a)

   Show that the liquid oscillates with simple harmonic motion and that the time period is $T = 2\pi\sqrt{\dfrac{L}{2g}}$.

   [3 marks]

   (b)

   The liquid is displaced by an amplitude of 15 mm and released. Calculate the time period and the maximum speed of the liquid.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A trolley of mass 0.400 kg rests on a horizontal, almost frictionless air track and is attached to a fixed support by a light spring of spring constant 25.0 N m⁻¹. The trolley is pulled 80 mm from equilibrium and released, so that it oscillates with simple harmonic motion. Later, the air supply is switched off and a card sail is fitted to the trolley.

   (a)

   The air supply is on, so the motion is undamped. Calculate the total energy of the oscillation, and the speed of the trolley when its displacement is 50 mm.

   [6 marks]

   (b)

   Describe how the kinetic energy, potential energy and total energy of the trolley vary with time while the air supply is on. Explain how the motion changes when the air is switched off and the sail is fitted, and what would be different if the damping were increased until the trolley just returned to equilibrium without oscillating.

   [6 marks]

   Total for question 4: 12 marks

End of questions