Lesson 11 of 11 · 16 min

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Must-know facts

17 facts

1Mass is the measure of inertia; heavier bodies are harder to start, stop or turn.
2F = dp/dt; for constant mass F = ma. 1 N = 1 kg m s⁻².
3Impulse = FΔt = Δp; unit N s, same as momentum.
4Rebound: Δp = m(v₁ + v₂) in magnitude because the velocity reverses.
5Action and reaction never cancel each other: the two act on different bodies.
6Weight of a book and the normal force on it are balancing forces on one body, not a third-law pair.
7Total momentum of an isolated system is conserved; recoil speed of a gun is (m_bullet/m_gun) × bullet speed.
8Static friction is self-adjusting: f_s ≤ μ_s N; it equals μ_s N only at the point of slipping.
9Kinetic friction f_k = μ_k N, and usually μ_k < μ_s.
10Limiting friction does not depend on the area of contact.
11Rolling friction is far smaller than sliding friction.
12Apparent weight in a lift = m(g + a) for upward acceleration and m(g − a) for downward acceleration; zero in free fall.
13Maximum speed on a level curve: v = √(μ_s R g), independent of mass.
14Optimum speed on a banked curve (no friction needed): v₀ = √(R g tan θ).
15Atwood machine: a = (m₁ − m₂)g/(m₁ + m₂), T = 2m₁m₂g/(m₁ + m₂).
16Tension is the same throughout a light string over a frictionless pulley.
17Acceleration at an instant depends only on the net force at that instant.

Common traps

Where marks are lost

Writing the friction on a body at rest as μ_s N whatever the applied force.

Static friction only matches the applied force. Compare the applied force with μ_s N first: if smaller, friction equals the applied force and the body does not move.

Subtracting speeds to find the change in momentum when a ball bounces back from a wall.

Velocity reverses, so Δv = v_final − v_initial = v₂ − (−v₁) = v₁ + v₂ in magnitude.

Cancelling an action-reaction pair because the forces are equal and opposite.

Third-law forces act on two different bodies. Only forces acting on the same body can be added in its equation of motion.

Assuming a lift moving upward always makes you feel heavier.

Apparent weight depends on the direction of acceleration. A lift moving up but slowing down accelerates downward, so the reading is m(g − a).

Adding centripetal force as an extra arrow on a free-body diagram.

Centripetal force is the net inward force produced by real forces (friction, tension, normal force, gravity). Draw only the real forces.

Taking the normal reaction as mg when a pulling force acts at an angle above the horizontal.

The upward component F sin θ reduces the normal force to mg − F sin θ; a downward push at an angle increases it.

Using the thrower's relative speed directly in a momentum-conservation equation.

Momentum must be written with velocities relative to the ground; convert a relative speed first, then conserve momentum.

Thinking friction always opposes the motion of a body.

Friction opposes relative motion between the surfaces in contact. It can point along a body's motion, as for a crate on an accelerating truck or a foot pushing back on the ground.

Formulas

16 to know

Linear momentum

p = mv

Vector along v; SI unit kg m s⁻¹.

Newton's second law

F = dp/dt = ma

F is the net external force; F = ma for constant mass.

Impulse

J = FΔt = Δp = m(v − u)

F is the average force during the short time Δt; unit N s.

Newton's third law

F_AB = −F_BA

The two forces act on different bodies.

Conservation of momentum (two bodies)

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Isolated system; velocities relative to the ground, with signs.

Recoil velocity

V = −(m/M)v

M, V for the gun; m, v for the bullet; starting from rest.

Equilibrium of a particle

ΣFx = 0, ΣFy = 0

Resolve forces along perpendicular axes.

Static friction

f_s ≤ μ_s N, (f_s)max = μ_s N

Self-adjusting up to the limiting value.

Kinetic friction

f_k = μ_k N

Usually μ_k < μ_s.

Spring force

F = −kx

Restoring force for small deformation x; k in N m⁻¹.

Apparent weight in a lift

R = m(g + a) upward acceleration; R = m(g − a) downward acceleration

R is the normal force (scale reading).

Centripetal force

F = mv²/R

Directed towards the centre of the circle.

Maximum speed on a level road

v_max = √(μ_s R g)

Independent of mass.

Maximum speed on a banked road

v_max = √[R g (μ_s + tan θ)/(1 − μ_s tan θ)]

θ is the banking angle.

Optimum speed on a banked road

v₀ = √(R g tan θ)

Friction not needed at this speed.

Two masses over a pulley (Atwood)

a = (m₁ − m₂)g/(m₁ + m₂), T = 2m₁m₂g/(m₁ + m₂)

Light inextensible string, frictionless light pulley, m₁ > m₂.

Key terms

14 terms

Inertia: The tendency of a body to keep its state of rest or uniform straight-line motion.
Inertial frame: A reference frame in which a body with no net force on it moves with constant velocity.
Linear momentum: Mass times velocity; the quantity whose rate of change equals the net force.
Impulse: The product of a large force and the short time it acts, equal to the change of momentum it produces.
Newton: The force that gives a 1 kg mass an acceleration of 1 m s⁻².
Normal reaction: The part of a contact force perpendicular to the surfaces in contact.
Tension: The pulling force a stretched string or rope exerts on whatever is attached to it.
Static friction: Friction that prevents sliding from starting; adjusts in size up to a limit.
Limiting friction: The largest static friction, μ_s N, reached just before sliding begins.
Kinetic friction: Friction that acts while surfaces slide over each other, μ_k N.
Rolling friction: The small resistance to a body rolling over a surface.
Centripetal force: The net inward force needed to keep a body moving in a circle.
Banking of roads: Raising the outer edge of a curved road so that the normal force helps provide the centripetal force.
Free-body diagram: A sketch of one body showing all the external forces acting on it.

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