Worksheet: Elastic Potential Energy

In this worksheet, we will practice finding the work done in stretching an elastic string or spring and finding the energy stored in an elastic string or spring.

Q1:

A ball of mass 𝑚 is attached to one end of a light elastic string of natural length 3𝑎 and modulus of elasticity 3𝑚𝑔, where 𝑔 is the acceleration due to gravity. The other end of the string is fixed at a point 𝑂 on the line where the base of a vertical wall meets a rough horizontal plane. The ball is held on the plane a distance of 15𝑎2 from 𝑂 in such a way that the string lies perpendicular to the wall. Given that the coefficient of friction between the ball and the plane is 34, find the speed of the ball 𝑣 at the moment it collides with the wall. Give your answer in terms of 𝑎 and 𝑔. If the ball rebounds from the wall with speed 2𝑣9, at what distance 𝑑 from 𝑂 will the ball finally come to rest?

  • A𝑣=34√21𝑎𝑔, 𝑑=𝑎9
  • B𝑣=3√𝑎𝑔, 𝑑=8𝑎27
  • C𝑣=3√𝑎𝑔, 𝑑=13𝑎27
  • D𝑣=34√26𝑎𝑔, 𝑑=8𝑎27
  • E𝑣=34√26𝑎𝑔, 𝑑=13𝑎27

Q2:

A light elastic spring has a natural length 1.4 m and modulus of elasticity 22 N. How much work is done in compressing the spring from a length of 1.3 m to a length of 1.2 m?

  • A1135 J
  • B3370 J
  • C11140 J
  • D1170 J
  • E33140 J

Q3:

One end of a light elastic string of natural length 3.9 m and modulus of elasticity 17.3 N is attached to a fixed point 𝐴 on a rough horizontal surface. A ball of mass 2.9 kg is attached to the other end of the string, and the ball is placed on the surface so that the string is taut but not stretched. The ball is then subjected to an impulse resulting in a velocity of 1.1 m/s directly away from 𝐴. Given that the coefficient of friction between the ball and the surface is 0.3, find the distance the ball travels before it is instantaneously at rest. Give your answer in meters, correct to two decimal places, taking 𝑔=9.8/ms.

Q4:

A particle 𝑃 of mass 𝑚 is attached to one end of a light elastic string of natural length 𝑙 and modulus of elasticity 2𝑚𝑔. The other end of the string is fixed at a point 𝐴 which is at a height of 4𝑙 above a horizontal floor. The particle 𝑃 is projected vertically downward from 𝐴 at a speed of 𝑢, where 𝑢=12𝑔𝑙. Given that when the particle hits the floor it rebounds at half the speed, find the maximum speed of 𝑃 during its motion.

  • A√30𝑔𝑙2
  • B√29𝑔𝑙2
  • C√15𝑔𝑙
  • D√58𝑔𝑙2
  • E√14𝑔𝑙

Q5:

A particle of mass 9𝑚 hangs at the end of a light elastic string of natural length 4𝑎. The string is connected to a horizontal ceiling, and has length of 9𝑎 at equilibrium. Find the elastic potential energy stored in the string. Consider the acceleration due to gravity to be 𝑔.

  • A81𝑚𝑔𝑎2
  • B45𝑚𝑔𝑎
  • C45𝑚𝑔𝑎4
  • D18𝑚𝑔𝑎
  • E45𝑚𝑔𝑎2

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