How does the period of a loaded spring change if it is replaced with a stiffer spring of twice the spring constant, carrying the same mass?

The period falls by a factor of √2, to about 71% of its original value, because T is inversely proportional to the square root of the spring constant.

Calculate the period of a 0.20 kg mass oscillating on a spring of spring constant 32 N m⁻¹.

T = 2π√(0.20/32) = 2π√(6.25 × 10⁻³) = 2π × 0.0791 = 0.50 s.

Calculate the spring constant of a spring on which a mass of 0.50 kg oscillates with a period of 0.99 s.

Rearranging gives k = 4π²m/T² = 39.5 × 0.50/0.980 = 19.7/0.980 = 20 N m⁻¹.

CCEA A-Level Physics 1210

Physical Quantities, Scalars and Vectors

Topic overview Physical quantities and SI base units Prefixes and standard form Scalars and vectors Resolving vectors and finding resultants Coplanar forces in equilibrium

Moments and Equilibrium

Topic overview Moment of a force Centre of gravity The principle of moments Determining unknown masses using moments

Linear Motion and Dynamics

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Newton's Laws of Motion

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Linear Momentum and Impulse

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Work, Energy and Power

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Current, Charge and Potential Difference

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Resistance and Resistivity

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Waves

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Wave-Particle Duality

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Astronomy

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Practical Skills and Data Analysis

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Thermal Physics

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Uniform Circular Motion

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Simple Harmonic Motion

Topic overview Defining simple harmonic motion and a = −ω²x Displacement, velocity and acceleration: x = A cos ωt The simple pendulum and T = 2π√(l/g)The loaded spiral spring and T = 2π√(m/k)SHM graphs and finding velocity from a gradient Free vibrations, forced vibrations and resonance Damping: light, critical and over-damping

The Nucleus and Radioactive Decay

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Nuclear Energy, Fission and Fusion

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Gravitational Fields

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Magnetic Fields

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Fundamental Particles

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Simple Harmonic Motion

The loaded spiral spring and T = 2π√(m/k)

3 things to know
Unit A2 1

Overview

Recall questions on the period equation with every symbol, why a mass on a spring performs simple harmonic motion, the effect of quadrupling the mass and of using a stiffer spring, worked period and spring-constant calculations, and what is plotted to determine the spring constant from a set of periods.

What you need to know

Answer each one before you reveal it.

Your recallNot started

Key term

State the equation for the period of a mass oscillating on a spiral spring and define every symbol.

Key term

Why does a mass on a spiral spring oscillate with simple harmonic motion?

Key term

How does the period of a loaded spring change if the mass hung on it is quadrupled?

Questions from this subtopic

Each one with the model answer underneath.

1
State the equation for the period of a mass oscillating on a spiral spring and define every symbol.
Model answer
T = 2π√(m/k), where T is the period in seconds, m the oscillating mass in kilograms and k the spring constant in N m⁻¹.
2
Why does a mass on a spiral spring oscillate with simple harmonic motion?
Model answer
The spring obeys Hooke's law, so the restoring force is proportional to the displacement from equilibrium and acts towards it, which is exactly the condition for simple harmonic motion.
3
How does the period of a loaded spring change if the mass hung on it is quadrupled?
Model answer
The period doubles, because T is proportional to the square root of the mass.
4
How does the period of a loaded spring change if it is replaced with a stiffer spring of twice the spring constant, carrying the same mass?
Model answer
The period falls by a factor of √2, to about 71% of its original value, because T is inversely proportional to the square root of the spring constant.
5
Calculate the period of a 0.20 kg mass oscillating on a spring of spring constant 32 N m⁻¹.
Model answer
T = 2π√(0.20/32) = 2π√(6.25 × 10⁻³) = 2π × 0.0791 = 0.50 s.
6
Calculate the spring constant of a spring on which a mass of 0.50 kg oscillates with a period of 0.99 s.
Model answer
Rearranging gives k = 4π²m/T² = 39.5 × 0.50/0.980 = 19.7/0.980 = 20 N m⁻¹.

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