Physics A Level
Chapter 16: Circular motion 16.7 The origins of centripetal forces
Physics A Level
Chapter 16: Circular motion 16.7 The origins of centripetal forces
It is useful to look at one or two situations where the physical origin of the centripetal force may not be immediately obvious. In each case, you will notice that the forces acting on the moving object are not balanced – there is a resultant force. An object moving along a circular path is not in equilibrium and the resultant force acting on it is the centripetal force.
1- A car cornering on a level road (Figure 16.13). Here, the road provides two forces. The force N is the normal contact force that balances the weight mg of the car–the car has no acceleration in the vertical direction.
The second force is the force of friction F between the tyres and the road surface. This is the unbalanced, centripetal force. If the road or tyres do not provide enough friction, the car will not go round the bend along the desired path. The friction between the tyres and the road provides the centripetal force necessary for the car’s circular motion.
2- A car cornering on a banked road (Figure 16.14a). Here, the normal contact force N has a horizontal component that can provide the centripetal force. The vertical component of N balances the car’s weight. Therefore:
$\begin{array}{l}
vertically\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,N\,\cos \theta = mg\\
horizontally\,\,\,\,\,\,\,\,\,\,\,\,\,\,N\,\sin \theta = \frac{{m{v^2}}}{r}
\end{array}$
where r is the radius of the circular corner and v is the car’s speed.
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If a car travels around the bend too slowly, it will tend to slide down the slope and friction will act up the slope to keep it on course (Figure 16.14b). If it travels too fast, it will tend to slide up the slope. If friction is insufficient, it will move up the slope and come off the road.
3- An aircraft banking (Figure 16.15a). To change direction, the pilot tips the aircraft’s wings. The vertical component of the lift force L on the wings balances the weight. The horizontal component of L provides the centripetal force.
4- A stone being whirled in a horizontal circle on the end of a string – this arrangement is known as a conical pendulum (Figure 16.15b). The vertical component of the tension T is equal to the weight of the stone. The horizontal component of the tension provides the centripetal force for the circular motion.
5- At the fairground (Figure 16.15c). As the cylinder spins, the floor drops away. Friction balances your weight. The normal contact force of the wall provides the centripetal force. You feel as though you are being pushed back against the wall; what you are feeling is the push of the wall on your back.
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Note that the three situations shown in Figures 16.14a, 16.15a and 16.15b are equivalent. The moving object’s weight acts downwards. The second force has a vertical component, which balances the weight, and a horizontal component, which provides the centripetal force.
Questions
19) Explain why it is impossible to whirl a bung around on the end of a string in such a way that the string remains perfectly horizontal.
20) Explain why an aircraft will tend to lose height when banking, unless the pilot increases its speed to provide more lift.
21) If you have ever been down a water-slide (a flume) (Figure 16.16) you will know that you tend to slide up the side as you go around a bend. Explain how this provides the centripetal force needed to push you around the bend. Explain why you slide higher if you are going faster.
EXAM-STYLE QUESTIONS
1) Which statement is correct? [1]
A: There is a resultant force on an object moving along a circular path at constant speed away from the centre of the circle causing it to be thrown outwards.
B: There is a resultant force on an object moving along a circular path at constant speed towards the centre of the circle causing it to be thrown outwards.
C: There is a resultant force on an object moving along a circular path at constant speed towards the centre of the circle causing it to move in the circle.
D: There is zero resultant force on an object moving along a circular path at constant speed because it is in equilibrium.
2) When ice-dancers spin, as shown in the diagram, the first dancer’s hand applies a centripetal force to the second dancer’s hand.
In which case is the centripetal force the greatest? [1]
| x / m | Speed of the female skater’s centre of $mass/m\,{s^{ - 1}}$ | |
| A | 0.45 | 9.0 |
| B | 0.45 | 10.0 |
| C | 0.50 | 9.0 |
| D | 0.50 | 10.0 |
3) a: Explain what is meant by a radian. [1]
b: A body moves round a circle at a constant speed and completes one revolution in $15 s$. Calculate the angular speed of the body. [2]
[Total: 3]
4) This diagram shows part of the track of a roller-coaster ride in which a truck loops the loop. When the truck is at the position shown, there is no reaction force between the wheels of the truck and the track. The diameter of the loop in the track is $8.0 m$.
a: Explain what provides the centripetal force to keep the truck moving in a circle. [1]
b: Given that the acceleration due to gravity g is $9.8\,m\,{s^{ - 2}}$, calculate the speed of the truck. [3]
[Total: 4]
5) This diagram shows a toy of mass $60 g$ placed on the edge of a rotating turntable. [1]
a: The radius of the turntable is $15.0 cm$. The turntable rotates, making 20 revolutions every minute. Calculate the resultant force acting on the toy. [3]
b: Explain why the toy falls off when the speed of the turntable is increased. [2]
[Total: 6]
6) One end of a string is secured to the ceiling and a metal ball of mass $50 g$ is tied to its other end. The ball is initially at rest in the vertical position. The ball is raised through a vertical height of $70 cm$, as shown. The ball is then released. It describes a circular arc as it passes through the vertical position.
The length of the string is $1.50 m$.
a: Ignoring the effects of air resistance, determine the speed v of the ball as it passes through the vertical position. [2]
b: Calculate the tension T in the string when the string is vertical. [3]
c: Explain why your answer to part b is not equal to the weight of the ball. [2]
[Total: 7]
7) A car is travelling round a bend when it hits a patch of oil. The car slides off the road onto the grass verge. Explain, using your understanding of circular motion, why the car came off the road.
[2]
8) This diagram shows an aeroplane banking to make a horizontal turn. The aeroplane is travelling at a speed of $75\,m\,{s^{ - 1}}$ and the radius of the turning circle is $800 m$.
a: Copy the diagram. On your copy, draw and label the forces acting on the aeroplane. [2]
b: Calculate the angle that the aeroplane makes with the horizontal. [4]
[Total: 6]
9) a: Explain what is meant by the term angular speed. [2]
b: This diagram shows a rubber bung, of mass $200 g$, on the end of a length of string being swung in a horizontal circle of radius $40 cm$. The string makes
an angle of ${56^ \circ }$ with the vertical.
Calculate:
i- the tension in the string [2]
ii- the angular speed of the bung [3]
iii- the time it takes to make one complete revolution. [1]
[Total: 8]
10) a: Explain what is meant by a centripetal acceleration. [2]
b: A teacher swings a bucket of water, of total mass $5.4 kg$, round in a vertical circle of diameter $1.8 m$.
i- Calculate the minimum speed that the bucket must be swung at so that the water remains in the bucket at the top of the circle. [3]
ii- Assuming that the speed remains constant, what will be the force on the teacher’s hand when the bucket is at the bottom of the circle? [2]
[Total: 7]
11) In training, military pilots are given various tests. One test puts them in a seat on the end of a large arm that is then spun round at a high speed, as shown.
a: Describe what the pilot will feel and relate this to the centripetal force. [3]
b: At top speed the pilot will experience a centripetal force equivalent to six times his own weight ($6 mg$).
i- Calculate the speed of the pilot in this test. [3]
ii- Calculate the number of revolutions of the pilot per minute. [2]
c: Suggest why it is necessary for pilots to be able to be able to withstand forces of this type. [2]
[Total: 10]
12) a: Show that in one revolution there are 2π radians. [2]
b: This diagram shows a centrifuge used to separate solid particles suspended in a liquid of lower density. The container is spun at a rate of 540 revolutions per minute.
i- Calculate the angular velocity of the container. [2]
ii- Calculate the centripetal force on a particle of mass $20 mg$ at the end of the test tube. [2]
c: An alternative method of separating the particles from the liquid is to allow them to settle to the bottom of a stationary container under gravity.
By comparing the forces involved, explain why the centrifuge is a more effective method of separating the mixture. [2]
[Total: 8]
SELF-EVALUATION CHECKLIST
After studying this chapter, complete a table like this:
| I can | See topic… | Needs more work | Almost there | Ready to move on |
| define the radian and use it as the unit of angular displacement | 16.2 | |||
| understand the concept of angular speed | 16.4 | |||
| recall and use the relationship angular speed where T is the time for one complete revolution | 16.4 | |||
| recall and use the relationship angular speed $v = \omega r$ | 16.4 | |||
| understand that the force on an object rotating round a circle is towards the centre of the circle and is called a centripetal force | 16.5 | |||
| recognise that the centripetal force is at right angles to the velocity of the object | 16.5 | |||
| recognise that the centripetal force causes centripetal acceleration | 16.5 | |||
| recognise that a constant centripetal force causes circular motion with constant angular speed | 16.5 | |||
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recall and use the formula: $a = \frac{{{v^2}}}{r} = r{\omega ^2}$ |
16.6 | |||
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recall and use the formula: $F = \frac{{m{v^2}}}{r} = mr{\omega ^2}$ |
16.6 |