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Further mechanics extends the principles of Year 1 to cover circular motion and oscillatory systems. These concepts are essential for understanding everything from satellite orbits to musical instruments, and from suspension bridges to radio tuning circuits.
When an object moves in a circle at constant speed, its velocity is constantly changing direction. This means the object is accelerating, even though its speed is constant. The acceleration is always directed towards the centre of the circle.
Key Definition: Angular velocity (ω) is the rate of change of angular displacement, measured in rad s⁻¹.
ω = 2π/T = 2πf = Δθ/Δt
where T is the period and f is the frequency.
The linear speed of an object moving in a circle of radius r is:
v = ωr
The acceleration is always directed towards the centre of the circle:
a = v²/r = ω²r
By Newton's second law, a resultant force must act towards the centre to maintain circular motion:
F = mv²/r = mω²r
This is the centripetal force. It is not a new type of force — it is provided by whatever force keeps the object on its circular path (e.g. gravity for a planet, tension in a string, friction on a road).
Key Definition: The centripetal force is the resultant force directed towards the centre of the circular path that maintains circular motion.
| Situation | What Provides the Centripetal Force |
|---|---|
| Satellite in orbit | Gravitational attraction |
| Car on a flat roundabout | Friction between tyres and road |
| Ball on a string (horizontal circle) | Horizontal component of tension |
| Electron in a magnetic field | Magnetic force (BQv) |
| Car on a banked road | Horizontal component of normal contact force |
Simple harmonic motion can be understood as the projection of uniform circular motion onto a diameter. If an object moves in a circle of radius A at angular velocity ω, its projection onto the x-axis has displacement x = A cos(ωt), which is exactly the equation for SHM.
A car of mass 1200 kg travels around a banked curve of radius 80 m. The road is banked at 15° to the horizontal. Calculate the speed at which the car can travel around the bend without any friction being required.
Described diagram — Forces on a car on a banked road: The car sits on a surface tilted at angle θ to the horizontal. The weight (mg) acts vertically downward. The normal contact force (N) acts perpendicular to the road surface. There is no friction. The horizontal component of N provides the centripetal force; the vertical component of N balances the weight.
Resolving vertically: N cos θ = mg → N = mg / cos θ
Resolving horizontally (towards centre): N sin θ = mv²/r
Dividing the horizontal equation by the vertical equation:
tan θ = v²/(rg)
v² = rg tan θ = 80 × 9.81 × tan 15° = 80 × 9.81 × 0.268 = 210.1
v = √210.1 = 14.5 m s⁻¹ (about 32 mph)
At this speed, no friction is needed. Below this speed, friction acts up the slope; above this speed, friction acts down the slope.
A ball of mass 0.20 kg is attached to a string of length 0.50 m and swung in a vertical circle. Calculate the minimum speed at the top of the circle for the string to remain taut.
At the top, both the weight and the tension act downward (towards the centre):
T + mg = mv²/r
For the string to just remain taut, T = 0:
mg = mv²/r → v² = rg = 0.50 × 9.81 = 4.905
v = √4.905 = 2.2 m s⁻¹
Exam Tip: In circular motion problems, always identify what provides the centripetal force. Draw a free body diagram and resolve forces towards the centre of the circle. Remember: centripetal force is not an additional force — it is the resultant of real forces.
Key Definition: Simple harmonic motion is oscillatory motion where the acceleration is always directed towards the equilibrium position and is proportional to the displacement from it: a = −ω²x.
The negative sign indicates that the acceleration is always in the opposite direction to the displacement (i.e. it is a restoring acceleration).
| Quantity | Equation |
|---|---|
| Displacement | x = A cos(ωt) or x = A sin(ωt) |
| Velocity | v = −Aω sin(ωt) or v = Aω cos(ωt) |
| Acceleration | a = −Aω² cos(ωt) or a = −ω²x |
| Maximum velocity | v_max = Aω (at equilibrium) |
| Maximum acceleration | a_max = Aω² (at extremes) |
| Velocity from displacement | v = ±ω√(A² − x²) |
| Angular frequency | ω = 2π/T = 2πf |
The velocity leads the displacement by π/2 (90°, or a quarter of a cycle). The acceleration leads the velocity by π/2, meaning the acceleration is π (180°) out of phase with the displacement (in antiphase).
Described diagram — Displacement-time graph for SHM: A smooth cosine curve oscillating between +A and −A. The period T is the time for one complete cycle. The curve starts at x = +A (if using x = A cos ωt) and passes through zero at t = T/4, reaches −A at t = T/2, passes through zero again at t = 3T/4, and returns to +A at t = T.
Described diagram — Velocity-time graph for SHM: A negative sine curve (if displacement is cosine). The velocity is zero when displacement is at its maximum (±A) and has its maximum magnitude (±Aω) when displacement is zero (at equilibrium). The velocity curve leads the displacement curve by a quarter cycle.
Described diagram — Acceleration-time graph for SHM: A negative cosine curve (if displacement is cosine) — it is the exact mirror image (reflection in the t-axis) of the displacement graph. When displacement is at +A, acceleration is at −Aω² (maximum magnitude, directed towards equilibrium). Acceleration is zero when displacement is zero.
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