A2 Circular motion questions

1. Two satellites of equal mass, S1 and S2, orbit the Earth. S1 is orbiting at a distance r from the

Earth’s centre at speed v. S2 orbits at a distance 2r from the Earth’s centre at speed 2

v

. The ratio of the centripetal force on S1 to the centripetal force on S2 is

A. 8

1

. B. 4

1

.

C. 4. D. 8.(1)

2. Which one of the following graphs best represents the variation of the kinetic energy, KE, and of the gravitational potential energy, GPE, of an orbiting satellite with its distance r from the centre of the Earth?

0

0

0

0

0

0

0

0

E n e r g y

E n e r g y

E n e r g y

E n e r g y

K E

K E

K E

K E

G P E

G P E

G P E

G P E

r

r

r

r

A .

C .

B .

D .

(1)

1

3. A particle P is moving in a circle with uniform speed. Which one of the following diagrams correctly shows the direction of the acceleration a and velocity v of the particle at one instant of time?

A . B .

C . D .

a

aa

a

v

v

v

v

P P

P P

(1)

4. Points P and Q are at distances R and 2R respectively from the centre X of a disc, as shown below.

Q

P 2 R

R X

The disc is rotating about an axis through X, normal to the plane of the disc. Point P has linear speed v and centripetal acceleration a. Which one of the following is correct for point Q?

Linear speed Centripetal acceleration

A. v a

B. v 2a

C. 2v 2a

D. 2v 4a

2

5. The centripetal force that causes a car to go round a bend in the road is provided by

A. the force produced by the car engine acting on the wheels.

B. the friction between the tyres and the road.

C. the weight of the car.

D. the force exerted by the driver on the steering wheel.(1)

6. The centripetal force F acting on a particle of mass m that is travelling with linear speed v along the arc of a circle of radius r is given by

A. F = .

2

mr

v

B. F = mv2r.

C. F = mr2v. D. F = .

2

r

mv

(1)

7. A point mass is moving in a horizontal circle with a velocity of constant magnitude v. At one particular time, the mass is at P. A short time later, the mass is at Q, as shown below.

P

Q

v

v

Which vector diagram correctly shows the change in velocity Δv of the mass during this time?

A .

C .

B .

D .

v

v v

v

v

v v

v

v

v v

v

(1)

3

8. A satellite of mass m and speed v orbits the Earth at a distance r from the centre of the Earth. The gravitational field strength due to the Earth at the satellite is equal to

A. r

v

. B. r

v 2

.

C. r

mv

. D. r

mv 2

.(1)

9. A brick is placed on the surface of a flat horizontal disc as shown in the diagram below. The discis rotating at constant speed about a vertical axis through its centre. The brick does not move relative to the disc.

Which of the diagrams below correctly represents the horizontal force or forces acting on the brick?

(1)

10. Which of the following expressions correctly relates the radius R of the circular orbit of a planetround a star to the period T of the orbit?

A. R3 T2 B.3

1

R T2

C. R2 T3 D.2

1

R T3

(1)

4

11. This question is about the kinematics and dynamics of circular motion.

(a) A car goes round a curve in a road at constant speed. Explain why, although its speed is constant, it is accelerating.

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................(2)

In the diagram below, a marble (small glass sphere) rolls down a track, the bottom part of whichhas been bent into a loop. The end A of the track, from which the marble is released, is at a height of 0.80 m above the ground. Point B is the lowest point and point C the highest point of the loop. The diameter of the loop is 0.35 m.

A

m a r b l e

g r o u n d B

C0 . 8 0 m

0 . 3 5 m

The mass of the marble is 0.050 kg. Friction forces and any gain in kinetic energy due to the

rotating of the marble can be ignored. The acceleration due to gravity, g = 10 ms–2.

Consider the marble when it is at point C.

(b) (i) On the diagram opposite, draw an arrow to show the direction of the resultant forceacting on the marble.

(1)

(ii) State the names of the two forces acting on the marble.

...........................................................................................................................

...........................................................................................................................(2)

5

(iii) Deduce that the speed of the marble is 3.0 ms–1.

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................(3)

(iv) Determine the resultant force acting on the marble and hence determine the reaction force of the track on the marble.

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................(4)

(Total 12 marks)

12. A satellite orbits the Earth at constant speed as shown below.

E a r t h

s a t e l l i t e

(a) Draw on the diagram

(i) an arrow labelled F to show the direction of the gravitational force of the Earth on the satellite.

(ii) an arrow labelled V to show the direction of the velocity of the satellite.(2)

6

(b) Although the speed of the satellite is constant, it is accelerating. Explain why it is accelerating.

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................(2)

(c) Discuss whether or not the gravitational force does work on the satellite.

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................(3)

(Total 7 marks)

13. This question is about atomic models. The diagram below (not to scale) shows a simple model of the hydrogen atom in which the electron orbits the proton in a circular path of radius R.

e l e c t r o nc h a r g e – e

p r o t o nc h a r g e + e

R

(a) On the diagram, draw an arrow to show the direction of

(i) the acceleration of the electron (label this A);(1)

(ii) the velocity of the electron (label this V).(1)

(b) State an expression for the magnitude of the electrostatic force F acting on the electron.

.....................................................................................................................................

.....................................................................................................................................(1)

7

(c) The orbital speed of the electron is 2.2 × 106 m s–1.

Deduce that the radius R of the orbit is 5.2 × 10–11 m.

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................(3)

(d) A more complex model of the atom suggests that the orbital radius can only take certain discrete values. This leads to the idea of discrete energy levels within the atom. Outline the evidence that supports the existence of discrete energy levels.

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................(3)

(Total 9 marks)

14. This question is about circular motion.

A linear spring of negligible mass requires a force of 18.0 N to cause its length to increase by 1.0 cm.

A sphere of mass 75.0 g is attached to one end of the spring. The distance between the centre of the sphere M and the other end P of the unstretched spring is 25.0 cm, as shown below.

2 5 . 0 c m

MP

The sphere is rotated at constant speed in a horizontal circle with centre P. The distance PM increases to 26.5 cm.

8

(a) Explain why the spring increases in length when the sphere is moving in a circle.

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................(2)

(b) Determine the speed of the sphere.

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................(4)

(Total 6 marks)

15. This question is about satellite motion.

A satellite of mass m orbits a planet of mass M and radius R as shown below. (The diagram is not to scale.)

p l a n e t m a s s M

R

x

s a t e l l i t e m a s s m

The radius of the circular orbit of the satellite is x. The planet may be assumed to behave as a point mass with its mass concentrated at its centre.

9

(a) Deduce that the linear speed v of the satellite in its orbit is given by the expression

v = x

GM

,

where G is the gravitational constant.

.....................................................................................................................................

.....................................................................................................................................

.....................................................................................................................................(2)

(b) (i) Derive expressions, in terms of m, G, M and x, for the kinetic energy of the satelliteand for the gravitational potential energy of the satellite.

Kinetic energy:

...........................................................................................................................

...........................................................................................................................

Gravitational potential energy:

...........................................................................................................................(2)

(ii) Deduce an expression for the total energy of the satellite.

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................(2)

The satellite is moved into an orbit closer to the planet where there is friction with the planet’s atmosphere.

(c) (i) State the effect of these frictional forces on the total energy of the satellite.

...........................................................................................................................(1)

(ii) Apply your equation in (b)(ii) to deduce that, as a result of this friction, the radius of the orbit will change continuously.

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................(2)

10

(iii) Describe the effect of this change in orbital radius on the speed of the satellite.

...........................................................................................................................(1)

(iv) The frictional forces will change as the orbit of the satellite changes. Suggest and explain the effect on the motion of the satellite of these changing frictional forces.

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................

...........................................................................................................................(3)

(Total 13 marks)

11