OCR A-Level Physics: Circular Motion, SHM and Gravitational Fields

A trolley of mass $m$m is attached between two identical springs on a smooth horizontal track. When displaced sideways and released, it oscillates with simple harmonic motion of amplitude $A$A.

Explain what is meant by the statement that the trolley moves with simple harmonic motion, and describe how the kinetic energy and the gravitational/elastic potential energy of the system change during one complete oscillation, stating clearly where in the motion each energy is a maximum and where each is zero.

(6 marks)

A car drives round a flat, horizontal bend that forms part of a circle. The only horizontal force available to keep the car on its circular path is the friction between the tyres and the road. The road is dry and the data are shown below.

Quantity	Value
Mass of car, $m$m	1200 kg
Radius of bend, $r$r	85 m
Maximum coefficient of friction, $\mu$μ	0.65
Gravitational field strength, $g$g	9.81 m s⁻²

(a) Calculate the maximum frictional force the road can provide on the car. (2 marks)

(b) Hence calculate the maximum speed at which the car can round the bend without skidding. (3 marks)

(c) The bend is resurfaced so that $\mu$μ falls to 0.45 in wet weather. State and explain what happens to the maximum safe speed. (1 mark)

A student investigates a simple pendulum by timing oscillations for several different lengths $L$L of the string. For each length she measures the period $T$T and calculates $T^2$T2. Her processed results are shown below.

Length $L$L / m	0.20	0.40	0.60	0.80	1.00
$T^2$T2 / s²	0.81	1.61	2.42	3.22	4.03

The period of a simple pendulum is given by $T = 2\pi\sqrt{\dfrac{L}{g}}$T=2πgL​​.

(a) Show that a graph of $T^2$T2 against $L$L should be a straight line through the origin, and state what its gradient represents. (2 marks)

(b) Use the first and last data points to determine the gradient of the graph. (1 mark)

(c) Hence determine a value for the gravitational field strength $g$g. (2 marks)

A small scientific probe is placed in a circular orbit around a distant planet of mass $M = 6.4 \times 10^{23} \ \text{kg}$M=6.4×1023 kg. The radius of the orbit, measured from the centre of the planet, is $r = 9.4 \times 10^{6} \ \text{m}$r=9.4×106 m. The gravitational constant is $G = 6.67 \times 10^{-11} \ \text{N m}^2\,\text{kg}^{-2}$G=6.67×10−11 N m2kg−2.

(a) By equating the gravitational force to the centripetal force, show that the orbital speed of the probe is given by $v = \sqrt{\dfrac{GM}{r}}$v=rGM​​, and calculate its value. (3 marks)

(b) Hence calculate the orbital period of the probe. (2 marks)

A mass of 0.25 kg is attached to the end of a vertical spring of spring constant 40 N m⁻¹. The mass is pulled down a short distance and released, so that it oscillates with simple harmonic motion of amplitude 0.080 m.

(a) Calculate the period of the oscillation. (2 marks)

(b) Calculate the maximum speed of the mass during its motion. (2 marks)

Gravitational fields are described by Newton's law of gravitation and by the field strength at a point.

(a) State Newton's law of gravitation in words, and give the corresponding equation, defining each symbol you use. (2 marks)

(b) Define the gravitational field strength at a point in a gravitational field. (1 mark)