Conical pendulumsAQA A-Level Further Maths: Subtopic test

1. 1
   A particle of mass $0.4$ kg is attached to a fixed point $O$ by a light inextensible string of length $0.8$ m. The particle moves in a horizontal circle below $O$ with constant speed, with the string taut and making an angle of $30^\circ$ with the downward vertical. Take $g=9.8$ m s⁻².

   (a)

   Find the tension in the string.

   [1 mark]

   • A$3.9$ N
   • B$7.8$ N
   • C$4.5$ N
   • D$2.3$ N

   (b)

   Find the radius of the circle.

   [1 mark]

   • A$0.40$ m
   • B$0.69$ m
   • C$0.80$ m
   • D$0.46$ m

   (c)

   Find the angular speed of the particle.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle of mass $0.6$ kg is attached to a fixed point $O$ by a light inextensible string of length $0.5$ m. It moves in a horizontal circle below $O$, with the string taut and making an angle $\theta$ with the downward vertical, where $\cos\theta=0.8$. Take $g=9.8$ m s⁻².

   (a)

   Find the tension in the string.

   [1 mark]

   • A$5.88$ N
   • B$9.80$ N
   • C$4.41$ N
   • D$7.35$ N

   (b)

   Find the speed of the particle.

   [1 mark]

   • A$2.2$ m s⁻¹
   • B$1.5$ m s⁻¹
   • C$2.0$ m s⁻¹
   • D$1.3$ m s⁻¹

   (c)

   Find the time taken for one complete revolution.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle of mass $m$ kg is attached to a fixed point $O$ by a light inextensible string of length $L$ m. It moves in a horizontal circle below $O$ with constant angular speed $\omega$ rad s⁻¹, with the string taut and making an angle $\theta$ with the downward vertical. Take $g=9.8$ m s⁻².

   (a)

   Show that $\cos\theta=\frac{g}{\omega^2L}$.

   [3 marks]

   (b)

   The string has length $0.9$ m.
   (i) Find $\omega$ when $\theta=60^\circ$.
   (ii) Explain why $\omega$ must exceed a certain value, and find this value.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $P$ of mass $0.6$ kg is attached to the ends of two light inextensible strings $AP$ and $BP$, each of length $0.5$ m. The point $A$ is vertically above $B$, with $AB=0.6$ m. The particle moves in a horizontal circle with constant angular speed $\omega$ rad s⁻¹ about the line $AB$, with both strings taut. Take $g=9.8$ m s⁻².

   (a)

   Find the tension in each string when $\omega=7$.

   [6 marks]

   (b)

   Find the least value of $\omega$ for which both strings are taut. Explain what happens if $\omega$ is smaller than this value.

   [6 marks]

   Total for question 4: 12 marks

End of questions