Newton's Second Law (F = ma)

The acceleration of an object is directly proportional to the resultant force acting on it and inversely proportional to its mass. The acceleration is in the direction of the resultant force.

$\boxed{F = m \times a}$

Where:

• $F$ = resultant force (in newtons, N)
• $m$ = mass (in kilograms, kg)
• $a$ = acceleration (in metres per second squared, m/s²)

Rearranging the Equation

$a = \frac{F}{m} \qquad m = \frac{F}{a}$

What the Equation Tells Us

1. Greater force → greater acceleration (for the same mass)

   • Pushing harder on a trolley makes it accelerate faster

2. Greater mass → smaller acceleration (for the same force)

   • A fully loaded lorry accelerates more slowly than an empty one with the same engine force

3. The direction of acceleration is the same as the direction of the resultant force

Defining the Newton

One newton (N) is defined as the force needed to give a mass of 1 kg an acceleration of 1 m/s²:

$1 \text{ N} = 1 \text{ kg} \times 1 \text{ m/s}^2 = 1 \text{ kg m/s}^2$

$F \propto a$ (at constant mass)

If you double the force, the acceleration doubles.

Graph: A straight line through the origin when plotting $a$ against $F$ (at constant $m$).

$a \propto \frac{1}{m}$ (at constant force)

If you double the mass, the acceleration halves.

Graph: A curve when plotting $a$ against $m$. A straight line through the origin when plotting $a$ against $1/m$.

1. Identify the resultant force (this is the net force, not just one of the forces)
2. Write the values you know: $F$, $m$, $a$
3. Rearrange $F = ma$ if necessary
4. Substitute the values
5. Calculate and give the answer with correct units

Aim

To investigate the effect of varying force and varying mass on the acceleration of an object.

Equipment

• Dynamics trolley
• Ramp or flat surface
• Pulley
• String
• Slotted masses (hanging weights)
• Light gates or ticker timer
• Balance

Method (Varying Force)

1. Set up a trolley on a flat, smooth surface connected by a string over a pulley to hanging masses
2. Keep the total mass constant (trolley + hanging masses)
3. Start with one hanging mass — release the trolley and measure its acceleration
4. Move one mass from the trolley to the hanging masses (keeps total mass the same)
5. Repeat for several different hanging masses
6. Plot a graph of acceleration (y-axis) against force (x-axis)

Expected Result (Varying Force)

A straight line through the origin — showing acceleration is directly proportional to force (when mass is constant).

Method (Varying Mass)

1. Keep the hanging mass (force) constant
2. Add masses to the trolley to change the total mass
3. For each mass, release the trolley and measure the acceleration
4. Plot a graph of acceleration (y-axis) against 1/mass (x-axis)

Expected Result (Varying Mass)

A straight line through the origin — showing acceleration is directly proportional to $1/m$ (inversely proportional to mass) when force is constant.

Common Sources of Error

• Friction: Compensate by slightly tilting the ramp until the trolley moves at constant speed when given a push
• Timing errors: Use light gates for more accurate measurements than manual timing
• Mass of string: Should be negligible compared to masses used

GCSE exams sometimes ask you to estimate forces. Here are some useful values:

• Newton's Second Law: $F = ma$ (resultant force = mass × acceleration)
• $F$ must be the resultant force, not an individual force
• Greater force → greater acceleration (for same mass)
• Greater mass → smaller acceleration (for same force)
• The newton is defined as the force to accelerate 1 kg at 1 m/s²
• Required practical: investigate how force and mass affect acceleration
• Can be combined with other equations ($W = mg$, SUVAT) for complex problems