What does \lambda represent?

The modulus of elasticity of the string, in newtons.

State the elastic potential energy of a stretched string.

\frac{\lambda x^2}{2l}=\frac12kx^2

Why is there a factor \frac12 in the elastic potential energy?

The tension rises from zero to T, so the work done is the average tension times the extension.

What is the extension?

Stretched length minus natural length.

Three energy forms in a conservation problem with a string?

Kinetic, gravitational potential and elastic potential energy.

Where is the speed greatest for a particle falling on a vertical string?

Where tension equals weight (resultant force zero), not at the lowest point.

What is the kinetic energy of a particle at the lowest point of its fall?

Zero: it is instantaneously at rest.

Hooke's law and elastic energyAQA A-Level Further Maths: Flashcards

Card 1 of 120 of 12 known

Question

State Hooke's law for an elastic string.

Tap or press Space to reveal

Tap card or press Space to flip

See all 12 cards

State Hooke's law for an elastic string.: $T=\frac{\lambda x}{l}$ , so tension is proportional to extension.
What does $\lambda$ represent?: The modulus of elasticity of the string, in newtons.
What is the stiffness $k$ in terms of $\lambda$ and $l$ ?: $k=\frac{\lambda}{l}$ , so $T=kx$ .
State the elastic potential energy of a stretched string.: $\frac{\lambda x^2}{2l}=\frac12kx^2$
Why is there a factor $\frac12$ in the elastic potential energy?: The tension rises from zero to $T$ , so the work done is the average tension times the extension.
What tension acts in a slack string?: None: it is zero.
A string with $l=1.2$ m, $\lambda=36$ N is stretched to 1.5 m. Find $T$ .: $\frac{36\times0.3}{1.2}=9$ N
What is the extension?: Stretched length minus natural length.
What is the condition for equilibrium of a particle hanging on a vertical string?: $\frac{\lambda x}{l}=mg$
Three energy forms in a conservation problem with a string?: Kinetic, gravitational potential and elastic potential energy.
Where is the speed greatest for a particle falling on a vertical string?: Where tension equals weight (resultant force zero), not at the lowest point.
What is the kinetic energy of a particle at the lowest point of its fall?: Zero: it is instantaneously at rest.

Exam questions on Hooke's law and elastic energy

A light elastic string has natural length 1.2 m and modulus of elasticity 36 N. One end is fixed to a point $A$ on a smooth horizontal table and a particle $P$ of mass 0.5 kg is attached to the other end. $P$ is held on the table with the string stretched to a total length of 1.5 m.
$P$ is released from rest. Find the speed of $P$ when the string reaches its natural length.2 marks
A particle of mass 2 kg hangs in equilibrium at the end of a light elastic string. The other end of the string is fixed to a point $O$ . The natural length of the string is 0.5 m and its extension in equilibrium is 0.25 m. Take $g=9.8\text{ m s}^{-2}$ .
The particle is pulled down until the extension of the string is 0.6 m and is then released from rest. Find the speed of the particle at the instant the string becomes slack.2 marks
A toy truck of mass 0.3 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 24 N. The other end of the string is fixed to a point $B$ on a smooth horizontal floor. The truck is pulled along the floor until it is 1.0 m from $B$ , with the string straight, and is then released from rest.
Find the tension in the string and the initial acceleration of the truck.3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).