A.2 · 13 · A.2 Forces and momentum · elastic restoring force, viscous drag, buoyancy, gravitational, electric and magnetic forces

Springs, viscous drag and field forces

The restoring force of a spring, drag on a small sphere in a fluid, and the three field forces: gravitational, electric and magnetic.

SL+HLHarder26 minFree

Hook

Pull a spring and it pulls back. Drop a bead into honey and it creeps. A charged balloon lifts your hair from a distance. Three very different pushes and pulls: how are they described?

Contact force

1 / 4

The elastic restoring force: Hooke’s law

F_H = −kx

k: spring constant (N m⁻¹)
x: extension or compression from the natural length (m)
the minus sign: the force is opposite to the displacement, so it always pulls the spring back to its natural length

This is a restoring force: it points back towards equilibrium. On a graph of force against extension a spring that obeys Hooke’s law is a straight line through the origin; the gradient is k, and the area under the line is the elastic energy stored, ½kx². Beyond the limit of proportionality the line curves and the law no longer holds.

Simulation

Force, extension and work

The idea: Stretch the spring and read F and the stored energy. Check the gradient is k.

Spring constant k40 N m⁻¹Extension x0.1 m

Force now, F = kx

4 N

Area = ½ × x × F

0.2 J

Elastic PE = ½kx²

0.2 J

The force grows as you stretch, so you cannot use force × distance with one force. The work is the area of the triangle under the line, and that gives ½kx². Double x and the energy goes up four times.

Check 1 of 5

A spring with k = 200 N m⁻¹ is stretched by 0.050 m. The force it exerts is…

Check 2 of 5

The minus sign in F_H = −kx means the spring force…

Check 3 of 5

In F_d = 6πηrv, doubling the speed of the sphere (same fluid, same sphere) makes the drag…

Check 4 of 5

Which force acts on a charged particle at rest in a magnetic field?

Check 5 of 5

An astronaut’s mass on the Moon compared with on Earth is…

Apply

A small steel ball falls at constant speed through glycerine. Name the three forces acting on it and explain why the speed becomes constant.

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Remember earlier lessons

From lesson A.2 · 3

F = kx describes a spring’s…

From lesson A.2 · 10

The buoyancy force equals…

Summary card

Key points

• Spring: F_H = −kx, restoring, straight line through the origin on an F–x graph.
• Small sphere in a fluid: F_d = 6πηrv, opposes the motion.
• Field forces act without contact: F_g = mg, F_e = qE, F_m = qvB sin θ.
• The magnetic force acts only on moving charges and is perpendicular to the velocity.

Formulas

F_H = −kx
F_d = 6πηrv
F_g = mg
F_e = qE

Key terms

Restoring force: a force that acts to return a body to equilibrium
Viscosity η: the property of a fluid that resists flow

Common errors

• Using 6πηrv for a fast or large object
• Calling weight a mass
• Forgetting that a magnetic force needs a moving charge

Checks solved: 0 of 5

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