Worksheet: Hooke's Law

In this worksheet, we will practice finding the force of stretching or compressing an elastic string or elastic spring using Hooke's law.

Q1:

A ball of mass 0.5 kg hangs in equilibrium from the ceiling by a light elastic spring of natural length 1.2 m and a modulus of elasticity 9.8 N. How much energy is stored in the spring? Take 𝑔=9.8/ms.

Q2:

An elastic string of natural length 4 m and modulus of elasticity 8 N is stretched to a length of 4.5 m. How much energy is stored in the stretched string?

Q3:

A ball of mass 1.8 kg is attached to one end of a light elastic string of natural length 2.4 m and modulus of elasticity 17.1 N. The other end of the string is fixed at a point 𝑂. The ball is released from rest at 𝑂. Taking 𝑔=9.8/ms, find how far below 𝑂 the ball reaches before coming instantaneously to rest.

Q4:

A particle of mass 4π‘š hangs in equilibrium at the end of an elastic string of natural length 6π‘Ž connected to a ceiling. The length of the string is then 8π‘Ž. Find the elastic potential energy stored in the string. Consider the acceleration due to gravity to be 𝑔.

  • A 1 2 π‘š 𝑔 π‘Ž
  • B 2 π‘š 𝑔 π‘Ž
  • C 1 6 π‘š 𝑔 π‘Ž
  • D 8 π‘š 𝑔 π‘Ž
  • E 4 π‘š 𝑔 π‘Ž

Q5:

A uniform rod 𝐴𝐡 of mass 3.5 kg and length 4 m is fixed at 𝐴, while its other end 𝐡 is connected by an elastic string, with a modulus of elasticity 11.1 N, to a point 𝐢, which lies at the same height as 𝐴, and 5.8 m from it. The system is in equilibrium when π‘šβˆ π΄π΅πΆ=90∘. Taking 𝑔=9.8/ms, find the tension in the elastic string when the system is in equilibrium, and hence find its natural length. Give your answers correct to two decimal places.

  • A 𝑇 = 1 1 . 8 3 N, 𝑙 = 1 . 3 4 m
  • B 𝑇 = 2 . 4 1 N, 𝑙 = 3 . 4 5 m
  • C 𝑇 = 2 3 . 6 6 N, 𝑙 = 1 . 3 4 m
  • D 𝑇 = 2 3 . 6 6 N, 𝑙 = 2 . 0 3 m
  • E 𝑇 = 1 1 . 8 3 N, 𝑙 = 2 . 0 3 m

Q6:

One end of a light elastic string is attached to a fixed point. A force of 3.8 N is applied to the other end of the string to stretch it. The natural length of the string is 2.8 m, and its modulus of elasticity is 26.6 N. Find the total length of the stretched string.

Q7:

A particle of mass 1 kg is attached to point 𝑂 on horizontal ceiling by a light elastic string of natural length 0.6 m and modulus of elasticity 18 N. The particle is held at a distance of 1.3 m directly below point 𝑂 and released from rest. Find the initial acceleration of the particle. Consider the acceleration due to gravity to be 9.8 m/s2.

Q8:

A bead of mass 742 g hangs from a ceiling by a light elastic string 𝐴𝐡 with a natural length of 2 m and a modulus of elasticity of 20 N. Given that the string is fixed to the ceiling at point 𝐴 and that the bead is moving in a horizontal circle such that the string is inclined at 14∘ to the vertical, find the angular speed of the bead in radians per second correct to one decimal place. Take 𝑔=9.8/ms.

Q9:

𝑂 is a fixed point on a rough plane, inclined at angle 𝛼 to the horizontal, where tan𝛼=0.65. An elastic string of natural length 2.8 m and modulus of elasticity 37.6 N is attached to 𝑂 at one end and a ball of mass 2.5 kg at the other end. Given that the ball was held at 𝑂 and then released from rest, coming to a stop again after moving 5.6 m down the plane, find the coefficient of friction between the ball and the plane to two decimal places. Take 𝑔=9.8/ms.

Q10:

A rod 𝐴𝐡 has mass 4π‘š and length 2π‘Ž. The rod rests horizontally in equilibrium, and is suspended from a fixed point 𝐢 by two light strings 𝐴𝐢 and 𝐡𝐢. 𝐴𝐢 makes an angle 𝛼 with the rod and is perpendicular to 𝐡𝐢, as shown in the figure. The string 𝐡𝐢 is elastic, with a natural length π‘Ž and a modulus of elasticity π‘˜π‘šπ‘”. Assume π‘˜ is a constant and 𝑔 is the acceleration due to gravity. Find the value of π‘˜.

Q11:

A ball with mass 2.3 kg hangs from the ceiling by a light elastic string with a natural length of 2.4 m and a modulus of elasticity of 29.4 N. Determine the extension of the string, considering the acceleration due to gravity to be 9.8 m/s2.

Q12:

A 4 kg body is hanging in equilibrium and suspended at 𝑃 by two light strings 𝐴𝑃 and 𝐡𝑃. The two strings are attached to two fixed points 𝐴 and 𝐡 on the same horizontal level. 𝐡𝑃 is inextensible, while 𝐴𝑃 is elastic with a natural length of 0.8 m and a modulus of elasticity of 9.8 N. 𝐴𝑃 makes an angle of 30∘ with 𝐴𝐡, and the two strings make a right angle at 𝑃. Find the length of 𝐡𝑃 and its tension 𝑇. Give both of your answers correct to one decimal place. Take 𝑔=9.8/ms.

  • A 𝐡 𝑃 = 0 . 9 m , 𝑇 = 1 9 . 6 N
  • B 𝐡 𝑃 = 2 . 8 m , 𝑇 = 3 3 . 9 N
  • C 𝐡 𝑃 = 1 . 4 m , 𝑇 = 3 3 . 9 N
  • D 𝐡 𝑃 = 1 . 8 m , 𝑇 = 3 9 . 2 N
  • E 𝐡 𝑃 = 1 . 4 m , 𝑇 = 1 9 . 6 N

Q13:

A light elastic spring with a modulus of elasticity of 20.4 N was fixed to the ceiling. The length of the spring when a ball of mass 0.4 kg was attached to its other end was 2 m. Find the length of the spring when the first ball is removed and another ball of mass 1.9 kg is attached to it, rounding your answer to 2 decimal places. Take 𝑔=9.8/ms.

Q14:

A light elastic spring sits vertically on a horizontal plane. A ball 𝐡 of mass π‘š is placed on the spring. When the system comes to rest, the spring has been compressed by a length of π‘Ž. The ball is now taken off the spring, and some time later it is dropped onto the spring from a height of 8π‘Ž9 above the top of the uncompressed spring. Find the spring’s maximum compression in the motion that follows in terms of π‘Ž.

  • A 2 π‘Ž 3
  • B 1 , 4 2 9 π‘Ž 6 4 9
  • C 3 π‘Ž 2
  • D 3 π‘Ž 8
  • E 8 π‘Ž 3

Q15:

The ends of a light spring of natural length 17𝑙 are fixed at points 𝐴 and 𝐡. The points lie at the same height and are a distance of 17𝑙 apart. A ball is attached to the midpoint of the spring, held there, and then released. When the ball comes instantaneously to rest, 𝐴, 𝐡, and the ball lie on the vertices of an equilateral triangle. Find the modulus of elasticity of the spring in terms of the mass of the ball, π‘š, and the acceleration due to gravity, 𝑔.

  • A √ 3 π‘š 𝑔
  • B 1 7 √ 3 3 π‘š 𝑔
  • C 1 7 √ 3 6 π‘š 𝑔
  • D 1 7 2 π‘š 𝑔
  • E 1 7 √ 3 π‘š 𝑔

Q16:

A light elastic string is attached to a horizontal ceiling. When a particle of mass 1 kg is attached to the other end of the string, it hangs 1 m below the ceiling at equilibrium. When the 1 kg particle is removed and a particle of mass 7 kg is attached to the same string, it hangs 6 m below the ceiling at equilibrium. Find the modulus of elasticity of the string, taking the acceleration due to gravity to be 9.8 m/s2

Q17:

A light elastic string has a length of 4 m when a mass of 1 kg is suspended from it. The same string has a length of 9 m when a mass of 5 kg is suspended from it. Find the natural length of the string. Consider the acceleration due to gravity to be 9.8 m/s2.

Q18:

Two springs, both of natural length 𝑙 and modulus of elasticity 3π‘šπ‘”, are joined end to end at point 𝐡. A particle of mass π‘š is attached to the point where the springs are joined. The free ends of the springs are fixed at 𝐴 and 𝐢, where 𝐴 is a distance 7𝑙 vertically above 𝐢. When the system is in equilibrium, how far below 𝐴 is the ball? Consider the acceleration due to gravity to be 𝑔.

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

Q19:

A light elastic spring has a modulus of elasticity 27 N. The spring has a length of 0.3 m when compressed by a force of 17 N. Determine the natural length of the spring.

Q20:

A particle 𝑃 with mass π‘š hangs in equilibrium at a point 𝐴 from a fixed point 𝑂 on the ceiling by a light inextensible string of length 𝐿, where 𝐴 is vertically below 𝑂. Suppose the particle is pushed such that its horizontal speed is 5βˆšπ‘”πΏ3, where 𝑔 is the acceleration due to gravity. Find 𝑃’s height above 𝐴 at the instant the string becomes slack.

  • A 5 9 𝐿 2 7
  • B 3 4 𝐿 2 7
  • C 1 7 𝐿 3
  • D 1 7 𝐿 9
  • E 7 𝐿 2 7

Q21:

One end of a light elastic spring is attached to a fixed point 𝑂 on a rough plane inclined at an angle 𝛼 to the horizontal, where tan𝛼=1.3. The other end of the spring is attached to a ball of mass 1.3 kg. The spring has a natural length of 1 m and modulus of elasticity of 11 N. The coefficient of friction between the ball and the plane is 0.3. Given that the ball was held at rest on the inclined plane at a point 0.6 m away from 𝑂 down the line of greatest slope of the plane and then it was released, find its initial acceleration to the nearest two decimal places as it moves down the slope. Take 𝑔=9.8/ms.

Q22:

Suppose that a light elastic string of natural length 1.7 m was stretched between two points, 𝐴 and 𝐡, on a smooth horizontal table, where 𝐴𝐡=3.4m. If the tension in the string is 23 N, find its modulus of elasticity.

Q23:

A light elastic string, 𝑃𝐴, has a natural length of 0.5 m and a modulus of elasticity of 29.4 N. Another light string, 𝑃𝐡, is inextensible. End 𝐴 of the elastic string and end 𝐡 of the inextensible string are attached to two fixed points on the same horizontal ceiling. The other ends of each string are joined together at point 𝑃 and attached to a particle of mass 4 kg. The particle hangs in equilibrium below 𝐴𝐡 such that 𝑃𝐴 makes an angle of 30∘ with 𝐴𝐡. 𝑃𝐴 is also perpendicular to 𝑃𝐡. Find the length of 𝑃𝐴, considering the acceleration due to gravity to be 9.8 m/s2.

  • A 1 3 m
  • B 7 1 2 m
  • C 2 3 m
  • D 5 3 m
  • E 5 6 m

Q24:

One end of a light string of natural length 0.9 m and modulus of elasticity 27 N is attached to a fixed point 𝐴. The other end of the string is attached to a particle of mass 3 kg. The particle is held at a point below 𝐴 and then released from rest. Find the length of the string when the particle reaches its maximum speed. Take 𝑔=9.8/ms.

Q25:

A small smooth object 𝑆 of mass 1.4 kg is attached to one end of a light elastic string of natural length 2.1 m and modulus of elasticity 15 N. The other end of the string is fixed at a point 𝐴 on a smooth horizontal table. The object was placed on the table so that the string was straight and at its natural length, and then it was projected directly away from 𝐴 at a speed of 3 m/s. How fast was 𝑆 going when it was a distance of 2.7 m away from 𝐴? Give your answer in meters per second correct to two decimal places.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.