Motion in a vertical circleEdexcel 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 $1$ m. The other end of the string is fixed at a point $O$. The particle is at rest at the lowest point and is then projected horizontally with speed $8$ m s$^{-1}$. It moves in a complete vertical circle. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the speed of the particle at the highest point.

   [1 mark]

   • A$6.66$ m s$^{-1}$
   • B$4.98$ m s$^{-1}$
   • C$24.8$ m s$^{-1}$
   • D$10.2$ m s$^{-1}$

   (b)

   Find the tension in the string immediately after projection.

   [1 mark]

   • A$27.1$ N
   • B$32.0$ N
   • C$36.9$ N
   • D$41.8$ N

   (c)

   Find the tension in the string when the particle is at the highest point.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A bead $B$ of mass $0.1$ kg is threaded on a smooth circular wire of radius $0.5$ m, fixed in a vertical plane with centre $O$. $B$ is projected from the lowest point of the wire with speed $5$ m s$^{-1}$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the speed of $B$ when it is level with $O$.

   [1 mark]

   • A$2.32$ m s$^{-1}$
   • B$4.48$ m s$^{-1}$
   • C$15.2$ m s$^{-1}$
   • D$3.90$ m s$^{-1}$

   (b)

   Find the least speed of projection from the lowest point for $B$ to reach the highest point of the wire.

   [1 mark]

   • A$4.43$ m s$^{-1}$
   • B$4.95$ m s$^{-1}$
   • C$3.13$ m s$^{-1}$
   • D$19.6$ m s$^{-1}$

   (c)

   Find the magnitude and direction of the force exerted on $B$ by the wire when $B$ is at the highest point of the wire.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A smooth sphere of radius $0.5$ m is fixed with its centre at $O$. A particle $P$ of mass $0.4$ kg is projected horizontally with speed $1$ m s$^{-1}$ from the highest point $A$ of the sphere, and moves on the outer surface. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the speed of $P$ when $OP$ makes an angle of $30^\circ$ with the upward vertical and $P$ is still on the sphere.

   [3 marks]

   (b)

   Find the angle that $OP$ makes with the upward vertical when $P$ leaves the surface of the sphere.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass $0.25$ kg is attached to one end of a light inextensible string of length $0.6$ m. The other end of the string is fixed at a point $O$. $P$ hangs at rest at the lowest point and is then projected horizontally with speed $4$ m s$^{-1}$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find (i) the speed of $P$ when the string is horizontal, (ii) the tension in the string at that instant, (iii) the least speed of projection from the lowest point that would allow $P$ to complete a vertical circle.

   [6 marks]

   (b)

   Show that the string becomes slack before $P$ reaches the highest point. Find the angle the string then makes with the horizontal, and the speed of $P$ at that instant.

   [6 marks]

   Total for question 4: 12 marks

End of questions