Horizontal circular track – friction only (no banking)

1 A railway engine of mass 50000 kg travels at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a horizontal circular track of radius 1250 m . Find the magnitude of the horizontal force on the engine.

2 \includegraphics[max width=\textwidth, alt={}, center]{a20a6641-d771-4c89-b40f-168a0c61f99d-2_456_871_1228_635} An aircraft flies horizontally at a constant speed of \(220 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Initially it is flying due east. On reaching a point \(A\) it flies in a circular arc from \(A\) to \(B\), taking 50 s . At \(B\) the aircraft is flying due south (see diagram).

1. Show that the radius of the arc is approximately 7000 m .
2. Find the magnitude of the acceleration of the aircraft while it is flying between \(A\) and \(B\).

1 A child, of mass 40 kg , moves at constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a fairground ride.
The path of the child is a circle of radius 4 metres.
Find the magnitude of the resultant force acting on the child.
Circle your answer.
[0pt] [1 mark]
6.3 N
50 N
130 N
250 N

6 A car of mass 1200 kg travels round a roundabout on a horizontal, circular path at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The radius of the circle is 50 metres. Assume that there is no resistance to the motion of the car and that the car can be modelled as a particle.

1. A friction force, directed towards the centre of the roundabout, acts on the car as it moves. Show that the magnitude of this friction force is 4704 N .
2. The coefficient of friction between the car and the road is \(\mu\). Show that \(\mu \geqslant 0.4\).

5 A car travels around a roundabout at a constant speed. The surface of the roundabout is horizontal. The car has mass 990 kg and the path of the car is a circular arc of radius 48 metres.
A simple model assumes that the car is a particle and the only horizontal force acting on it as it travels around the roundabout is friction. On a dry day typical values of friction, \(F\), between the surface of the roundabout and the tyres of the car are $$7300 \mathrm {~N} \leq F \leq 9200 \mathrm {~N}$$ 5

1. Using this model calculate a safe speed limit, in miles per hour, for the car as it travels around the roundabout. Explain your reasoning fully.
   Note that there are 1600 metres in one mile.
   5
2. Gary assumes that on a wet day typical values for friction, \(F\), are $$5400 \mathrm {~N} \leq F \leq 10000 \mathrm {~N}$$ Comment on the validity of Gary's revised assumption.

4 A car of mass 1400 kg drives around a horizontal circular bend of radius 60 metres.
The car has a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) on the bend.
Calculate the magnitude of the resultant force acting on the car.
[0pt] [2 marks] \(5 \quad\) A region bounded by the curve with equation \(y = 4 - x ^ { 2 }\), the \(x\)-axis and the \(y\)-axis is shown below. \includegraphics[max width=\textwidth, alt={}, center]{cd0d239b-ab92-4d17-9cb8-45722e2894cb-04_641_380_408_831} The region is rotated through \(360 ^ { \circ }\) around the \(x\)-axis to create a uniform solid.

Aliya, whose mass is \(m\) kg, is playing rounders. She rounds the first base at a speed of \(v\) ms\(^{-1}\), making the turn on a horizontal circular path of radius \(r\) m.

1. Write down, in terms of \(m\), \(v\) and \(r\), the magnitude of the horizontal force acting on her. [1 mark]
2. Show that if she continues on the same circular path, the reaction force exerted on her by the ground must act at an angle \(\theta\) to the vertical, where \(\tan \theta = \frac{v^2}{gr}\). [6 marks]

A cyclist in a road race is travelling around a bend on a horizontal circular path of radius 15 metres and is prevented from skidding by a frictional force. The frictional force has a maximum value of 500 newtons. The total mass of the cyclist and his cycle is 75 kg Assume that the cyclist travels at a constant speed.

1. Work out the greatest speed, in km h\(^{-1}\), at which the cyclist can travel around the bend. [4 marks]
2. With reference to the surface of the road, describe one limitation of the model. [1 mark]