Hooke's law and elastic energyAQA A-Level Further Maths: Subtopic test

1. 1
   A light elastic string has natural length 1.2 m and modulus of elasticity 36 N. One end is fixed to a point $A$ on a smooth horizontal table and a particle $P$ of mass 0.5 kg is attached to the other end. $P$ is held on the table with the string stretched to a total length of 1.5 m.

   (a)

   Find the tension in the string.

   [1 mark]

   • A$7.2$ N
   • B$45$ N
   • C$9$ N
   • D$10.8$ N

   (b)

   Find the elastic potential energy stored in the string.

   [1 mark]

   • A$1.08$ J
   • B$2.7$ J
   • C$4.5$ J
   • D$1.35$ J

   (c)

   $P$ is released from rest. Find the speed of $P$ when the string reaches its natural length.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle of mass 2 kg hangs in equilibrium at the end of a light elastic string. The other end of the string is fixed to a point $O$. The natural length of the string is 0.5 m and its extension in equilibrium is 0.25 m. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the modulus of elasticity of the string.

   [1 mark]

   • A$19.6$ N
   • B$39.2$ N
   • C$58.8$ N
   • D$9.8$ N

   (b)

   Find the elastic potential energy stored in the string in the equilibrium position.

   [1 mark]

   • A$2.45$ J
   • B$4.9$ J
   • C$9.8$ J
   • D$19.6$ J

   (c)

   The particle is pulled down until the extension of the string is 0.6 m and is then released from rest. Find the speed of the particle at the instant the string becomes slack.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A toy truck of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N. The other end of the string is fixed to a point $B$ on a smooth horizontal floor. The truck is pulled along the floor until it is 1.0 m from $B$, with the string straight, and is then released from rest.

   (a)

   Find the tension in the string and the initial acceleration of the truck.

   [3 marks]

   (b)

   Find the speed of the truck when the string becomes slack.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle $Q$ of mass 1.5 kg is attached to one end of a light elastic string of natural length 2 m and modulus of elasticity 49 N. The other end of the string is fixed to a point $O$. $Q$ is released from rest at $O$ and falls vertically. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the greatest extension of the string.

   [6 marks]

   (b)

   Explain why the speed of $Q$ is greatest when the tension in the string equals the weight of $Q$, and find the greatest speed of $Q$.

   [6 marks]

   Total for question 4: 12 marks

End of questions