Mechanics 3 (M3): Motion in a circleEdexcel International A Level Further Maths: Topic test

1. 1
   A small insect sits on a horizontal turntable at a distance $0.12$ m from the axis of rotation. The turntable rotates at a constant $45$ revolutions per minute.

   (a)

   Find the angular speed of the turntable, in rad s$^{-1}$.

   [1 mark]

   • A$0.75$
   • B$4.71$
   • C$283$
   • D$1.5$

   (b)

   Find the speed of the insect, in m s$^{-1}$.

   [1 mark]

   • A$0.09$
   • B$0.18$
   • C$33.9$
   • D$0.565$

   (c)

   Find the magnitude of the acceleration of the insect, and state its direction.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   An aeroplane of mass $8000$ kg flies at a constant speed of $100$ m s$^{-1}$ in a horizontal circle of radius $2000$ m. The lift force $L$ acts perpendicular to the wings, and the wings are banked at an angle $\theta$ to the horizontal. Take $g=9.8$ m s$^{-2}$ and ignore air resistance.

   (a)

   Find the magnitude of the acceleration of the aeroplane, in m s$^{-2}$.

   [1 mark]

   • A$5$
   • B$0.05$
   • C$20$
   • D$0.51$

   (b)

   Find $\tan\theta$.

   [1 mark]

   • A$1.96$
   • B$0.255$
   • C$0.510$
   • D$0.0051$

   (c)

   Find the magnitude of the lift force $L$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A small ball of mass $0.2$ kg moves on the smooth inner surface of a fixed vertical circular track of radius $0.9$ m and centre $O$. The ball is projected from the lowest point $A$ of the track with speed $7$ m s$^{-1}$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the speed of the ball at the highest point $B$ of the track.

   [3 marks]

   (b)

   Find the magnitude of the normal reaction between the ball and the track at $B$, and hence show that the ball does stay in contact with the track at $B$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass $0.4$ kg is attached to two light inextensible strings $AP$ and $BP$, of lengths $1.3$ m and $0.5$ m respectively. The other ends of the strings are attached to fixed points $A$ and $B$, with $B$ vertically below $A$ and $AB=1.2$ m. The particle moves in a horizontal circle with constant angular speed, with both strings taut. Take $g=9.8$ m s$^{-2}$.

   (a)

   The angular speed of $P$ is $4$ rad s$^{-1}$. Find the tension in each string.

   [6 marks]

   (b)

   The angular speed of $P$ is gradually reduced. Find the angular speed at which $BP$ becomes slack, and the time for one revolution at that instant.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A pendulum bob $B$ of mass $0.5$ kg is attached to one end of a light inextensible string of length $1.5$ m. The other end of the string is fixed at $O$. The bob is released from rest when the string is taut and makes an angle of $60^\circ$ with the downward vertical. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the speed of $B$ when it passes through the lowest point of its path, in m s$^{-1}$.

   [1 mark]

   • A$5.42$
   • B$14.7$
   • C$3.83$
   • D$6.64$

   (b)

   Find the tension in the string when $B$ is at its lowest point, in N.

   [1 mark]

   • A$9.8$
   • B$4.9$
   • C$14.7$
   • D$12.25$

   (c)

   Find the speed of $B$ when the string makes an angle of $30^\circ$ with the downward vertical.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A car of mass $900$ kg, modelled as a particle, moves at a constant speed of $12$ m s$^{-1}$ around a circular roundabout of radius $40$ m on a horizontal road.

   (a)

   Find the angular speed of the car about the centre of the roundabout, in rad s$^{-1}$.

   [1 mark]

   • A$3.33$
   • B$480$
   • C$3.6$
   • D$0.3$

   (b)

   Find the time taken for one complete lap, in seconds.

   [1 mark]

   • A$10.5$
   • B$20.9$
   • C$0.0478$
   • D$3.33$

   (c)

   Find the horizontal resultant force on the car, and state its direction.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A smooth hollow hemispherical bowl of internal radius $0.5$ m is fixed with its rim horizontal and uppermost. A particle $P$ of mass $0.2$ kg moves with constant speed in a horizontal circle of radius $0.3$ m on the inner surface of the bowl. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the magnitude of the normal reaction of the bowl on $P$.

   [3 marks]

   (b)

   Find the angular speed of $P$ and the time it takes to complete one circle.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A humpback bridge has a profile that, at its highest point, is an arc of a vertical circle of radius $25$ m. A car of mass $1200$ kg, modelled as a particle, travels over the bridge. Take $g=9.8$ m s$^{-2}$.

   (a)

   The car passes the highest point at a constant speed of $10$ m s$^{-1}$. Find the normal reaction of the bridge on the car at this point, and find the greatest speed at which the car can pass the highest point without leaving the surface of the bridge.

   [6 marks]

   (b)

   The car now passes the highest point at $10$ m s$^{-1}$ with the engine off and moves on the smooth surface of the bridge. Find the angle that the radius to the car makes with the upward vertical at the point where the car leaves the surface of the bridge.

   [6 marks]

   Total for question 8: 12 marks

End of questions