Worksheet: Hooke’s Law

A particle of mass hangs in equilibrium at the end of an elastic string of natural length connected to a ceiling. The length of the string is then . Find the elastic potential energy stored in the string. Consider the acceleration due to gravity to be .

A uniform rod of mass 3.5 kg and length 4 m is fixed at , while its other end is connected by an elastic string, with a modulus of elasticity 11.1 N, to a point , which lies at the same height as , and 5.8 m from it. The system is in equilibrium when . Taking , 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.

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.

A particle of mass 1 kg is attached to point 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 and released from rest. Find the initial acceleration of the particle. Consider the acceleration due to gravity to be 9.8 m/s².

A bead of mass 742 g hangs from a ceiling by a light elastic string 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 and that the bead is moving in a horizontal circle such that the string is inclined at to the vertical, find the angular speed of the bead in radians per second correct to one decimal place. Take .

is a fixed point on a rough plane, inclined at angle to the horizontal, where . An elastic string of natural length 2.8 m and modulus of elasticity 37.6 N is attached to at one end and a ball of mass 2.5 kg at the other end. Given that the ball was held at 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 .

A rod has mass and length . The rod rests horizontally in equilibrium, and is suspended from a fixed point by two light strings and . makes an angle with the rod and is perpendicular to , as shown in the figure. The string is elastic, with a natural length and a modulus of elasticity . Assume is a constant and is the acceleration due to gravity. Find the value of .

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.

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

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 .

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

The ends of a light spring of natural length are fixed at points and . The points lie at the same height and are a distance of apart. A ball is attached to the midpoint of the spring, held there, and then released. When the ball comes instantaneously to rest, , , 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, , and the acceleration due to gravity, .

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

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.

Two springs, both of natural length and modulus of elasticity , are joined end to end at point . A particle of mass is attached to the point where the springs are joined. The free ends of the springs are fixed at and , where is a distance vertically above . When the system is in equilibrium, how far below is the ball? Consider the acceleration due to gravity to be .

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.

A particle with mass hangs in equilibrium at a point from a fixed point on the ceiling by a light inextensible string of length , where is vertically below . Suppose the particle is pushed such that its horizontal speed is , where is the acceleration due to gravity. Find ’s height above at the instant the string becomes slack.

One end of a light elastic spring is attached to a fixed point on a rough plane inclined at an angle to the horizontal, where . 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 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 .

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

A light elastic string, , has a natural length of 0.5 m and a modulus of elasticity of 29.4 N. Another light string, , is inextensible. End of the elastic string and end 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 and attached to a particle of mass 4 kg. The particle hangs in equilibrium below such that makes an angle of with . is also perpendicular to . Find the length of , considering the acceleration due to gravity to be 9.8 m/s².

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

A small smooth object 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 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 at a speed of 3 m/s. How fast was going when it was a distance of 2.7 m away from ? Give your answer in meters per second correct to two decimal places.