Elastic strings and springs and elastic energy

Section FM1-3

2 spec leafs

Notes and three levels of exam-style practice for each registered specification leaf in this section.

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FM1-3.1

Elastic strings and springs. Hooke's law.

For a spring obeying Hooke's law, the tension has magnitude $T=kx$ , where $k$ is the spring constant and $x$ is the extension.
For an elastic string of natural length $l$ and modulus $\lambda$ , Hooke's law is $T=\lambda x/l$ while the string is taut.
Extension is stretched length minus natural length. A string becomes slack rather than exerting a compressive force when this value is non-positive.
Keep spring constant and modulus distinct: $k$ has units $\text{N m}^{-1}$ , whereas $\lambda$ has units $\text{N}$ .

Tier 1 · Easy

2 marks

ORIGINAL

A light spring has spring constant $80\,\text{N m}^{-1}$ and is extended by $0.06\,\text{m}$ . Find its tension.

Tier 2 · Standard

3 marks

ORIGINAL

An elastic string has natural length $1.5\,\text{m}$ and modulus $120\,\text{N}$ . It is stretched to length $1.8\,\text{m}$ . Calculate its tension.

Tier 3 · Hard

4 marks

ORIGINAL

A $1.5\,\text{kg}$ particle hangs in equilibrium from a vertical elastic string of natural length $0.8\,\text{m}$ and modulus $49\,\text{N}$ . Taking $g=9.8\,\text{m s}^{-2}$ , determine the extension and the equilibrium length of the string.

FM1-3.2

Energy stored in an elastic string or spring.

A spring of constant $k$ stores elastic energy $\tfrac12kx^2$ when extended or compressed by $x$ .
A taut elastic string of natural length $l$ and modulus $\lambda$ stores energy $\lambda x^2/(2l)$ at extension $x$ .
Elastic energy is the area under the tension-extension graph, so it is quadratic in extension rather than equal to final tension times extension.
In an energy equation, include elastic energy only while the string is taut, and measure extension from the natural length at each position.

Tier 1 · Easy

2 marks

ORIGINAL

A spring of constant $200\,\text{N m}^{-1}$ is compressed by $0.08\,\text{m}$ . Calculate the elastic energy stored.

Tier 2 · Standard

3 marks

ORIGINAL

An elastic string has natural length $1.2\,\text{m}$ and modulus $60\,\text{N}$ . Find the energy stored when its length is $1.5\,\text{m}$ .

Tier 3 · Hard

5 marks

ORIGINAL

A $2\,\text{kg}$ particle is attached to the lower end of a vertical elastic string with natural length $1\,\text{m}$ and modulus $49\,\text{N}$ . It is released from rest when the string is at natural length. Taking $g=9.8\,\text{m s}^{-2}$ , find its speed after descending $0.6\,\text{m}$ .