Dynamics: Newton's Second Law

Kinematics tells us how objects move. Dynamics tells us why. At the heart of dynamics is Newton's second law — arguably the single most important equation in A-Level Physics. This lesson covers OCR Module 3.2.1 (Newton's laws of motion) and 3.2.2 (non-linear motion including drag), and lays the groundwork for momentum, energy and every later mechanics topic.

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Newton's Three Laws — A Brief Recap

• First Law: A body remains at rest or moves with constant velocity unless acted on by a resultant external force. (This defines the concept of inertia — the tendency of matter to resist changes in its state of motion.)
• Second Law: The rate of change of momentum of a body is proportional to the resultant force applied, and occurs in the direction of that force. For constant mass this reduces to F = ma.
• Third Law: For every action there is an equal and opposite reaction; forces always come in pairs acting on different bodies.

OCR examiners like to test the precise wording of these laws. Memorise them word-for-word.

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Newton's Second Law — Full Statement

The most general statement is:

F = dp/dt

where p = mv is momentum. For constant mass:

dp/dt = m (dv/dt) = ma

so:

F = ma

• F is the resultant (net) force in newtons (N).
• m is mass in kilograms (kg).
• a is acceleration in metres per second squared (m s⁻²).

One newton is defined as the resultant force that gives a 1 kg mass an acceleration of 1 m s⁻². The equation therefore defines the newton.

Common Exam Mistake: Using the applied force rather than the resultant force. Always find the vector sum of all forces acting on the body before substituting into F = ma.

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Weight and Mass

Mass is a scalar measure of the amount of matter in a body — unchanged anywhere in the universe. Weight is a vector force, the gravitational pull of the Earth (or other body) on the mass:

W = mg

• W in newtons, m in kg, g in m s⁻² (or equivalently N kg⁻¹).
• g is both the free-fall acceleration and the gravitational field strength; these are numerically identical by virtue of F = ma.
• On Earth g ≈ 9.81 N kg⁻¹; on the Moon g ≈ 1.62 N kg⁻¹; in deep space g ≈ 0.

A 70 kg astronaut has the same mass on Earth and on the Moon, but a weight of 687 N on Earth versus 113 N on the Moon. Confusing the two is a classic exam pitfall.

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Worked Example 1 — Lift Accelerating Upward

A person of mass 65 kg stands on bathroom scales in a lift. The lift accelerates upwards at 2.0 m s⁻². What reading (in newtons) do the scales show?

Let R be the normal force from the scales (this is what they measure). Taking upwards positive:

R − mg = ma R = m(g + a) = 65 × (9.81 + 2.0) = 65 × 11.81 = 768 N

Compared with the static reading (mg = 637 N), the person feels heavier because the scales must provide extra force to accelerate them upward. In free-fall the lift's acceleration would be a = −9.81 m s⁻² and R = 0 — the person would feel weightless.

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Worked Example 2 — Towing a Trailer

A car of mass 1200 kg tows a trailer of mass 500 kg. The engine provides a driving force of 3600 N. Resistance forces on car and trailer are 200 N and 150 N respectively.

(a) Find the acceleration. (b) Find the tension in the tow-bar.

(a) Consider the whole system:

• Total mass = 1200 + 500 = 1700 kg
• Resultant force = 3600 − 200 − 150 = 3250 N
• a = F / m = 3250 / 1700 = 1.91 m s⁻²

(b) Consider only the trailer:

• Forces on trailer: tension T forward, 150 N resistance backward.
• Newton II: T − 150 = 500 × 1.91 = 956
• T = 1106 N ≈ 1110 N

Cross-check: consider only the car.

• 3600 − T − 200 = 1200 × 1.91
• 3400 − T = 2294
• T = 1106 N ✓

The cross-check is a good habit — consistent answers mean consistent physics.

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Drag and Resistance Forces