Angular speed and circular motionAQA A-Level Further Maths: Subtopic test

1. 1
   A particle moves in a horizontal circle of radius 0.6 m with a constant speed of $3\text{ m s}^{-1}$.

   (a)

   Find the angular speed of the particle.

   [1 mark]

   • A$1.8\text{ rad s}^{-1}$
   • B$0.2\text{ rad s}^{-1}$
   • C$5\text{ rad s}^{-1}$
   • D$8.33\text{ rad s}^{-1}$

   (b)

   Find the magnitude of the acceleration of the particle.

   [1 mark]

   • A$15\text{ m s}^{-2}$
   • B$7.5\text{ m s}^{-2}$
   • C$5.4\text{ m s}^{-2}$
   • D$25\text{ m s}^{-2}$

   (c)

   Find the number of revolutions the particle makes per minute.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A fairground ride rotates at a constant 12 revolutions per minute. A rider sits at a distance of 4.5 m from the axis of rotation.

   (a)

   Find the angular speed of the ride in radians per second.

   [1 mark]

   • A$12\text{ rad s}^{-1}$
   • B$0.2\text{ rad s}^{-1}$
   • C$75.4\text{ rad s}^{-1}$
   • D$1.26\text{ rad s}^{-1}$

   (b)

   Find the speed of the rider.

   [1 mark]

   • A$0.9\text{ m s}^{-1}$
   • B$5.65\text{ m s}^{-1}$
   • C$339\text{ m s}^{-1}$
   • D$7.11\text{ m s}^{-1}$

   (c)

   Find the magnitude and direction of the acceleration of the rider.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A car of mass 900 kg travels at a constant speed of $15\text{ m s}^{-1}$ around a circular bend of radius 50 m.

   (a)

   Find the acceleration of the car and the magnitude of the resultant horizontal force acting on it.

   [3 marks]

   (b)

   The bend is a quarter of a circle (a turn of $90^\circ$). Find the angular speed of the car and the time it takes to travel round the bend.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two small objects, $P$ and $Q$, rest on a horizontal turntable at distances 0.4 m and 0.9 m from its centre. The turntable rotates at a constant 30 revolutions per minute and both objects move with it without slipping.

   (a)

   (i) Find the angular speed of the turntable in radians per second.
   (ii) Find the speed of $P$ and the speed of $Q$.
   (iii) Find the magnitude of the acceleration of $P$ and of $Q$.

   [6 marks]

   (b)

   Friction can provide an acceleration of at most $12\text{ m s}^{-2}$ for either object. The turntable's rate of rotation is gradually increased. Find the greatest rate of rotation, in revolutions per minute, at which $Q$ does not slip, and, by considering $P$ at that rate, explain why $Q$ slips before $P$.

   [6 marks]

   Total for question 4: 12 marks

End of questions