Horizontal circular motionEdexcel International A Level Further Maths: Subtopic test

1. 1
   A particle of mass $0.5$ kg is attached to one end of a light inextensible string of length $2$ m. The other end of the string is fixed to a point $O$ on a smooth horizontal table. The particle moves on the table in a circle with centre $O$, at a constant speed of $4$ m s$^{-1}$.

   (a)

   Find the tension in the string.

   [1 mark]

   • A$4$ N
   • B$8$ N
   • C$16$ N
   • D$1$ N

   (b)

   Find the angular speed of the particle.

   [1 mark]

   • A$8$ rad s$^{-1}$
   • B$0.5$ rad s$^{-1}$
   • C$32$ rad s$^{-1}$
   • D$2$ rad s$^{-1}$

   (c)

   The speed of the particle is increased until the tension in the string is three times its original value. Find the new speed.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle of mass $0.4$ kg is attached to one end of a light inextensible string of length $2.45$ m. The other end of the string is fixed. The particle moves in a horizontal circle with constant angular speed, with the string inclined at an angle $\theta$ to the vertical, where $\cos\theta=0.8$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the tension in the string.

   [1 mark]

   • A$3.14$ N
   • B$3.92$ N
   • C$4.9$ N
   • D$6.53$ N

   (b)

   Find the radius of the circle in which the particle moves.

   [1 mark]

   • A$1.96$ m
   • B$1.47$ m
   • C$1.84$ m
   • D$1.23$ m

   (c)

   Find the angular speed of the particle.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A car of mass $800$ kg is modelled as a particle moving in a horizontal circle of radius $60$ m on a road banked at an angle $\alpha$ to the horizontal, where $\tan\alpha=\frac14$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the speed at which the car can travel round the bend with no tendency to slip sideways.

   [3 marks]

   (b)

   The car travels round the bend at $15$ m s$^{-1}$ without slipping sideways. Find the least possible value of the coefficient of friction between the tyres and the road.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass $0.5$ kg is attached to one end of a light elastic string of natural length $0.8$ m and modulus of elasticity $40$ N. The other end of the string is fixed to a point $O$ on a smooth horizontal table. $P$ moves on the table in a horizontal circle with centre $O$, with the string stretched.

   (a)

   The string has length $1.0$ m. Find (i) the tension in the string, (ii) the speed of $P$, (iii) the time taken for $P$ to complete one revolution.

   [6 marks]

   (b)

   $P$ now moves with a constant angular speed of $5$ rad s$^{-1}$. Find (i) the length of the string, (ii) the tension in the string, (iii) the speed of $P$.

   [6 marks]

   Total for question 4: 12 marks

End of questions