Circular motion in a vertical planeAQA A-Level Further Maths: Subtopic test

1. 1
   A particle of mass $0.3$ kg is attached to one end of a light inextensible string of length $0.8$ m. The other end of the string is fixed at $O$. The particle moves in a vertical circle with centre $O$ and its speed at the lowest point $A$ is $7$ m s⁻¹. Take $g=9.8$ m s⁻².

   (a)

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

   [1 mark]

   • A$18.4$ N
   • B$15.4$ N
   • C$2.94$ N
   • D$21.3$ N

   (b)

   Find the speed of the particle at the highest point of the circle.

   [1 mark]

   • A$17.6$ m s⁻¹
   • B$4.2$ m s⁻¹
   • C$5.8$ m s⁻¹
   • D$1.4$ m s⁻¹

   (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 small bead $B$ of mass $0.05$ kg is threaded on a smooth circular wire of radius $0.4$ m, fixed in a vertical plane with centre $O$. The bead is projected from the lowest point of the wire with speed $4$ m s⁻¹. Take $g=9.8$ m s⁻².

   (a)

   Find the speed of the bead at the highest point of the wire.

   [1 mark]

   • A$0.57$ m s⁻¹
   • B$0.32$ m s⁻¹
   • C$2.9$ m s⁻¹
   • D$4.0$ m s⁻¹

   (b)

   Which condition must hold at the highest point for the bead to complete the circle?

   [1 mark]

   • A$v^2\ge gr$, as for a string
   • B$u\ge\sqrt{5gr}$ at the bottom
   • C$v\ge0$, because the wire can push or pull the bead
   • Dthe reaction must be zero

   (c)

   Find the least speed at the lowest point for which the bead reaches the highest point of the wire.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle of mass $0.2$ 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 particle hangs at rest at the lowest point and is then given a horizontal speed $u$ m s⁻¹. Take $g=9.8$ m s⁻².

   (a)

   Find the least value of $u$ for which the particle completes the vertical circle with the string always taut.

   [3 marks]

   (b)

   The particle is projected with $u=7$. Find the angle between the string and the horizontal at the instant the string becomes slack.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass $0.4$ kg moves on the smooth inner surface of a fixed circular track of radius $0.5$ m in a vertical plane, with centre $O$. The particle is projected from the lowest point of the track with speed $u$ m s⁻¹. It stays in contact with the track while the normal reaction is not negative. Take $g=9.8$ m s⁻².

   (a)

   The particle is projected with $u=6$.
   (i) Find its speed when $OP$ makes an angle of $60^\circ$ with the downward vertical.
   (ii) Find the normal reaction on $P$ at this position.

   [6 marks]

   (b)

   Find the least value of $u$ for which $P$ completes the circle without leaving the track. For this value of $u$, find the normal reaction at the lowest point.

   [6 marks]

   Total for question 4: 12 marks

End of questions