Newton's Laws of Motion

An object remains at rest or moves with constant velocity unless acted upon by a resultant force.

At A-Level depth:

• Defines inertial reference frames — frames where the law holds
• An object in equilibrium ($\sum \vec{F} = 0$) has zero acceleration
• Constant velocity includes both constant speed AND constant direction

$\boxed{\vec{F}_{\text{net}} = m\vec{a}}$

More generally (momentum form): $\vec{F} = \frac{d\vec{p}}{dt} = \frac{d(m\vec{v})}{dt}$

For constant mass: $F = ma$

Vector equation: Apply separately in each direction.

When object A exerts a force on object B, object B exerts an equal and opposite force on object A.

The two forces:

• Are equal in magnitude
• Opposite in direction
• Act on different objects
• Are the same type of force

Atwood Machine (Two Masses on a Pulley)

Masses $m_1$ (heavier) and $m_2$ connected by a light, inextensible string over a frictionless, massless pulley.

For $m_1$ (going down): m_1 g - T = m_1 a \tag{1}

For $m_2$ (going up): T - m_2 g = m_2 a \tag{2}

Add equations (1) and (2): $m_1 g - m_2 g = (m_1 + m_2)a$ $\boxed{a = \frac{(m_1 - m_2)g}{m_1 + m_2}}$

$\boxed{T = \frac{2m_1 m_2 g}{m_1 + m_2}}$

Body on a Table Connected to Hanging Mass

Mass $m_1$ on a frictionless table, connected by a string over the edge to hanging mass $m_2$.

For $m_1$ (horizontal): $T = m_1 a$

For $m_2$ (vertical): $m_2 g - T = m_2 a$

Adding: $m_2 g = (m_1 + m_2)a$ $a = \frac{m_2 g}{m_1 + m_2}$

For a mass $m$ on a slope at angle $\theta$:

Resolve weight along and perpendicular to the slope:

• Along slope (down): $mg\sin\theta$
• Perpendicular to slope: $mg\cos\theta$

Normal reaction: $N = mg\cos\theta$

Without friction: $ma = mg\sin\theta$ → $a = g\sin\theta$

With friction ($\mu$): $ma = mg\sin\theta - \mu mg\cos\theta$ $a = g(\sin\theta - \mu\cos\theta)$

The object accelerates down if $\tan\theta > \mu$.

• N1: No resultant force → no acceleration
• N2: $F = ma$ (apply in each direction independently)
• N3: Equal and opposite forces on different objects
• Connected bodies: Write $F = ma$ for each body; tension is internal force
• Pulleys: $a = (m_1-m_2)g/(m_1+m_2)$
• Inclined planes: Resolve weight parallel and perpendicular to slope