Vertical circular motionEdexcel A-Level Further Maths: Subtopic test

1. 1
   A particle $P$ 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 at a point $O$. $P$ moves in a vertical circle with centre $O$, with the string taut. At the lowest point $A$ of the circle, $P$ has speed 12 m s⁻¹. Take $g=9.8$ m s⁻².

   (a)

   Find the speed of $P$ at the highest point of the circle.

   [1 mark]

   • A$65.6$ m s⁻¹
   • B$10.2$ m s⁻¹
   • C$8.10$ m s⁻¹
   • D$4.57$ m s⁻¹

   (b)

   Find the tension in the string when $P$ is at the highest point.

   [1 mark]

   • A$11.5$ N
   • B$21.3$ N
   • C$16.4$ N
   • D$23.0$ N

   (c)

   Find the tension in the string when $P$ is at $A$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A smooth circular wire of radius 0.9 m is fixed in a vertical plane with centre $O$. A small bead $B$ of mass 0.2 kg is threaded on the wire and is projected from the lowest point $A$ of the wire with speed $u$ m s⁻¹. Take $g=9.8$ m s⁻².

   (a)

   What is the least value of $u$ for which $B$ completes the circle?

   [1 mark]

   • A$6.64$
   • B$4.20$
   • C$2.97$
   • D$5.94$

   (b)

   Given that $u=7$, find the speed of $B$ when $OB$ is horizontal.

   [1 mark]

   • A$3.70$ m s⁻¹
   • B$5.6$ m s⁻¹
   • C$6.34$ m s⁻¹
   • D$31.4$ m s⁻¹

   (c)

   Given that $u=7$, find the magnitude of the force exerted by the wire on $B$ when $OB$ is horizontal.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ of mass 0.4 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 $O$. $P$ is at rest at the point $A$ vertically below $O$ and is projected horizontally with speed 4 m s⁻¹. $P$ then moves along a circular path until the string becomes slack. Take $g=9.8$ m s⁻².

   (a)

   Find the tension in the string when $OP$ is horizontal.

   [3 marks]

   (b)

   The string becomes slack when $OP$ makes an angle $\alpha$ above the horizontal. Find $\alpha$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass 0.6 kg moves on the smooth inner surface of a fixed hollow sphere of radius 1.5 m and centre $O$. $P$ is projected horizontally from the lowest point $A$ of the sphere with speed 8 m s⁻¹. Take $g=9.8$ m s⁻².

   (a)

   Find, at the instant when $OP$ is horizontal, (i) the speed of $P$, (ii) the normal reaction on $P$, (iii) the radial and tangential components of the acceleration of $P$.

   [6 marks]

   (b)

   $P$ leaves the surface when $OP$ makes an angle $\theta$ above the horizontal. Find $\theta$ and the speed of $P$ at that instant.

   [6 marks]

   Total for question 4: 12 marks

End of questions