Worksheet: Elastic Potential Energy and Conservation of Energy

In this worksheet, we will practice solving problems that include elastic strings and springs by using the work–energy principle and the conservation of energy principle.

Q1:

A spring is fixed at one end and hangs vertically. Its lower end is pulled 7 cm down from its equilibrium position and released. It performs 20 oscillations every second, and the amplitude of the oscillations decreases by 1 1 % each second. Find a function that models 𝐷 , the displacement of the end of the spring from its equilibrium position, in terms of 𝑑 , the time in seconds after it is released.

  • A 𝐷 ( 𝑑 ) = 7 ( 0 . 1 1 ) ( 4 0 πœ‹ 𝑑 )  c o s
  • B 𝐷 ( 𝑑 ) = 2 0 ( 0 . 1 1 ) ( 7 πœ‹ 𝑑 )  c o s
  • C 𝐷 ( 𝑑 ) = 2 0 ( 0 . 8 9 ) ( 7 πœ‹ 𝑑 )  c o s
  • D 𝐷 ( 𝑑 ) = 7 ( 0 . 8 9 ) ( 2 0 πœ‹ 𝑑 )  c o s
  • E 𝐷 ( 𝑑 ) = 7 ( 0 . 8 9 ) ( 4 0 πœ‹ 𝑑 )  c o s

Q2:

A spring is fixed at one end and hangs vertically. Its lower end is pulled 32 cm down from its equilibrium position and released. It performs 8 oscillations every second, and the amplitude of the oscillations decreases by 5 0 % each second. How long does it take for the amplitude of its oscillations to decrease to 0.1 cm? Give your answer to three decimal places.

  • A 6.315 seconds
  • B 8.315 seconds
  • C 9.944 seconds
  • D 8.322 seconds
  • E 9.944 seconds

Q3:

A spring is fixed at one end and hangs vertically. Its lower end is pulled 17 cm down from its equilibrium position and released. It performs 14 oscillations every second, and, after 3 seconds, the amplitude of the oscillations is 13 cm. Find a function that models 𝐷 , the displacement of the end of the spring from its equilibrium position, in terms of 𝑑 , the time in seconds after it was released.

  • A 𝐷 ( 𝑑 ) = 1 7 ( 0 . 9 1 4 5 ) ( 1 4 πœ‹ 𝑑 )  c o s
  • B 𝐷 ( 𝑑 ) = 1 7 ( 0 . 9 1 4 5 ) ( 2 8 πœ‹ 𝑑 )  c o s
  • C 𝐷 ( 𝑑 ) = 1 7 ( 1 . 0 9 3 5 ) ( 2 8 πœ‹ 𝑑 )  c o s
  • D 𝐷 ( 𝑑 ) = 1 7 ( 1 . 0 9 3 5 ) ( 1 4 πœ‹ 𝑑 )  c o s
  • E 𝐷 ( 𝑑 ) = 1 4 ( 0 . 9 1 4 5 ) ( 1 7 πœ‹ 𝑑 )  c o s

Q4:

A particle of mass 3 kg is attached to end 𝑃 of a light elastic spring 𝑃 𝑄 of natural length 0.5 m and modulus of elasticity 41 N. The spring rests on a smooth horizontal plane with end 𝑄 fixed. The particle is held at rest and then released. Find the initial acceleration of the particle if 𝑃 𝑄 = 1 . 1 m initially. Take 𝑔 = 9 . 8 / m s  .

Q5:

Determine the elastic potential energy stored in an elastic spring of natural length 5 m and a modulus of elasticity 13.4 N, which was compressed to a length of 4 m.

Q6:

A bowl of mass 𝑀 hangs in equilibrium from the ceiling by a light elastic spring of natural length 0.8 m and modulus of elasticity 7 N. Taking 𝑔 = 9 . 8 / m s  , find the distance the bowl moves when a mass of 1 kg is gently placed in it.

  • A 1 3 2 5 m
  • B 8 3 5 m
  • C 1 4 2 5 m
  • D 2 6 2 5 m
  • E 4 3 5 m

Q7:

A long spring is formed by joining a light spring of natural length 3.9 m and modulus of elasticity 18.2 N to another light spring of natural length 2.8 m and modulus of elasticity 19.6 N. Find the tension in the combined spring when it is stretched to a length of 8.5 m.

Q8:

A ball of mass 0.6 kg hangs in equilibrium from a light spring of natural length 2.5 m and modulus of elasticity 9 N. The ball is then pulled vertically downwards and released from rest at a point 0.8 m below its equilibrium position. Taking 𝑔 = 9 . 8 / m s  , find the acceleration the ball experiences immediately after being released.

Q9:

A spring is fixed at one end and hangs vertically. Its lower end is pulled 19 cm down from its equilibrium position and released. It performs 13 oscillations every second and, after 4 seconds, the amplitude decreases to 14 cm. Find a function that models 𝐷 , the displacement of the end of the spring from its equilibrium position, in terms of 𝑑 , the time in seconds after it is released.

  • A 𝐷 ( 𝑑 ) = 1 9 ( 1 . 0 7 9 3 ) ( 2 6 πœ‹ 𝑑 )  c o s
  • B 𝐷 ( 𝑑 ) = 1 9 ( 0 . 9 2 6 5 ) ( 1 3 πœ‹ 𝑑 )  c o s
  • C 𝐷 ( 𝑑 ) = 1 9 ( 1 . 0 7 9 3 ) ( 1 3 πœ‹ 𝑑 )  c o s
  • D 𝐷 ( 𝑑 ) = 1 3 ( 0 . 9 2 6 5 ) ( 1 7 πœ‹ 𝑑 )  c o s
  • E 𝐷 ( 𝑑 ) = 1 9 ( 0 . 9 2 6 5 ) ( 2 6 πœ‹ 𝑑 )  c o s

Q10:

On a smooth horizontal surface, a light elastic string of natural length 1.2 m and modulus of elasticity 9 N is fixed at a point 𝐴 on that surface. At its other end, a ball of mass 2.7 kg is attached. If the ball was held at point 𝐴 and then projected horizontally with a speed of 2.2 m/s, how far does it travel before coming to an instantaneous rest?

Q11:

One end of a light elastic spring of natural length 3.5 m and modulus of elasticity 24.5 N is attached to a fixed point, 𝐴 , whereas the other end is attached to a particle of mass 1.9 kg. The particle is held at a point 𝐡 , which is 2.2 m vertically below 𝐴 . What distance does the particle fall before it comes to rest if it is projected vertically downwards from 𝐡 with a speed of 0.7 m/s? Consider the acceleration due to gravity to be 9.8 m/s2, and give your answer correct to two decimal places.

Q12:

A particle of mass 3.7 kg is attached to one end of a light elastic string of natural length 1.7 m and modulus of elasticity 291.1 N. The other end of the string is attached to a fixed point 𝑂 . Taking 𝑔 = 9 . 8 / m s  , find the maximum speed of the particle when it is released from rest at 𝑂 . Round the answer to two decimal places.

Q13:

One end of a light elastic spring of natural length 1.6 m and modulus of elasticity 14 N is fixed to a point 𝑂 on top of a table. A ball of mass 2 kg is attached to the other end of the spring. The ball is held 0.7 m above 𝑂 so that the spring is vertical and then released. Taking 𝑔 = 9 . 8 / m s  , find the magnitude and direction of the acceleration the ball experiences immediately after being released. Give the magnitude in meters per second squared correct to two decimal places.

  • A 5.86 m/s2, downward
  • B 2.94 m/s2, upward
  • C 13.74 m/s2, upward
  • D 4.94 m/s2, downward
  • E 6.74 m/s2, downward

Q14:

A ball, with centre 𝐴 and mass π‘š , is attached to one end of a light elastic string of natural length 𝑙 and modulus of elasticity 2 π‘š 𝑔 , where 𝑔 is the acceleration due to gravity. The other end of the string is fixed at a point 𝑂 on top of a rough horizontal surface. The ball was placed on the surface so that the string was stretched to a length of 7 𝑙 5 . Given that the ball remains at rest at this point and is in limiting equilibrium, determine the coefficient of friction πœ‡ between the ball and the surface. The ball is now moved such that the string is stretched to a length of 𝐴 𝑂 = 2 𝑙 , and then released from rest. Find the total distance, 𝑑 , that the ball will travel before coming to rest.

  • A πœ‡ = 2 5 , 𝑑 = 5 𝑙 2
  • B πœ‡ = 4 1 5 , 𝑑 = 5 𝑙 2
  • C πœ‡ = 4 5 , 𝑑 = 5 𝑙 9
  • D πœ‡ = 4 5 , 𝑑 = 5 𝑙 4
  • E πœ‡ = 2 5 , 𝑑 = 5 𝑙 1 1

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