Further Mechanics 1: Elastic strings and springsEdexcel A-Level Further Maths: Topic test

1. 1
   A light elastic spring has natural length $0.50$ m and modulus of elasticity $49$ N. Its upper end is fixed to a point on a ceiling and a particle of mass $2$ kg is attached to its lower end. The particle hangs at rest in equilibrium. Take $g=9.8$ m s$^{-2}$.

   (a)

   What is the tension in the spring when the particle is in equilibrium?

   [1 mark]

   • A$9.8$ N
   • B$19.6$ N
   • C$2.0$ N
   • D$196$ N

   (b)

   What is the extension of the spring when the particle is in equilibrium?

   [1 mark]

   • A$0.4$ m
   • B$0.5$ m
   • C$0.7$ m
   • D$0.2$ m

   (c)

   Find the elastic potential energy stored in the spring when the particle hangs in equilibrium.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A light elastic string has natural length $1.5$ m. When a force of $12$ N is applied to stretch it, the string has a length of $1.8$ m. The string obeys Hooke's law.

   (a)

   What is the extension of the string when the force of $12$ N is applied?

   [1 mark]

   • A$1.8$ m
   • B$3.3$ m
   • C$0.3$ m
   • D$1.5$ m

   (b)

   What is the modulus of elasticity of the string?

   [1 mark]

   • A$60$ N
   • B$72$ N
   • C$40$ N
   • D$12$ N

   (c)

   Find the tension in the string when its length is $2.1$ m.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A light spring has its upper end fixed to a ceiling and hangs vertically. When a particle of mass $1.5$ kg is attached to the lower end and hangs at rest, the length of the spring is $0.9$ m. When the particle is replaced by one of mass $2$ kg, the length of the spring at equilibrium is $1.0$ m. The spring has natural length $l$ metres and modulus of elasticity $\lambda$ newtons. Take $g=9.8$ m s$^{-2}$.

   (a)

   Show that $\frac{\lambda}{l}=49$.

   [3 marks]

   (b)

   Hence find the natural length $l$ and the modulus of elasticity $\lambda$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A bungee jumper of mass $50$ kg is modelled as a particle. One end of a light elastic rope, of natural length $20$ m and modulus of elasticity $5880$ N, is attached to the jumper and the other end is fixed to a point $O$ on a bridge. The jumper steps off the bridge from rest at $O$ and falls vertically. Air resistance may be ignored. Take $g=9.8$ m s$^{-2}$.

   (a)

   Show that the greatest extension of the rope is $10$ m.

   [6 marks]

   (b)

   Find the greatest tension in the rope, and use the work-energy principle to find the speed of the jumper when the extension of the rope is $5$ m.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A dart gun uses a light spring of natural length $0.15$ m and modulus of elasticity $30$ N. The spring is compressed to a length of $0.10$ m and then released, projecting a dart of mass $10$ g along a smooth horizontal barrel. The dart leaves the spring when the spring reaches its natural length.

   (a)

   What is the elastic potential energy stored in the compressed spring?

   [1 mark]

   • A$0.25$ J
   • B$0.5$ J
   • C$1.0$ J
   • D$0.0375$ J

   (b)

   What is the speed of the dart when it leaves the spring?

   [1 mark]

   • A$50$ m s$^{-1}$
   • B$5.00$ m s$^{-1}$
   • C$7.07$ m s$^{-1}$
   • D$22.4$ m s$^{-1}$

   (c)

   Find the compression at which the energy stored in the spring would be $0.5$ J.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A trolley of mass $2$ kg rests on a smooth horizontal track. It is attached to one end of a light elastic spring of natural length $0.5$ m and modulus of elasticity $20$ N. The other end of the spring is fixed to a point on the track. The trolley is pulled along the track until the spring is stretched by $0.2$ m, and is then released from rest.

   (a)

   What is the tension in the spring at the moment of release?

   [1 mark]

   • A$4$ N
   • B$0.8$ N
   • C$20$ N
   • D$8$ N

   (b)

   What is the speed of the trolley when the spring reaches its natural length?

   [1 mark]

   • A$0.800$ m s$^{-1}$
   • B$0.894$ m s$^{-1}$
   • C$1.26$ m s$^{-1}$
   • D$0.400$ m s$^{-1}$

   (c)

   Find the acceleration of the trolley at the moment of release.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A particle of mass $3$ kg is on a smooth plane inclined at $30^\circ$ to the horizontal. It is attached to one end of a light elastic string of natural length $1.0$ m and modulus of elasticity $98$ N. The other end of the string is fixed to the plane at the highest point of the line of greatest slope, and the string lies along the line of greatest slope. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the extension of the string when the particle is in equilibrium on the plane.

   [3 marks]

   (b)

   The particle is released from rest at the point where the string is just taut. Use an energy method to find the greatest extension of the string.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A particle of mass $0.8$ kg is attached to one end of a light elastic string of natural length $1.0$ m and modulus of elasticity $49$ N. The other end of the string is fixed to a point $O$ on a ceiling and the particle hangs vertically below $O$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the extension of the string when the particle hangs in equilibrium. The particle is then pulled vertically down until the extension of the string is $0.4$ m and is released from rest. Find the initial acceleration of the particle.

   [6 marks]

   (b)

   Find the speed of the particle at the instant the string becomes slack, and the further height through which the particle rises above that point.

   [6 marks]

   Total for question 8: 12 marks

End of questions