Newton's Second Law — F=ma Explained with Worked Examples

Definition

The acceleration 'a' of an object in the direction of a resultant force F,
is directly proportional to the magnitude of the imposed force and
inversely proportional to the mass m of the object.

See below illustration regarding the two scenarios when the forces are unbalanced:
newton-law-2

Newton's second law is about the relationship between: force, mass, and acceleration.

1 Newton 1 kg • m/s²

Fact Checking about Force and Mass

What happens when a force F acts on a mass m?
newton-second

(a) The force F produces an acceleration a.
(b) If the mass m remains the same and the force F is doubled, the acceleration a is doubled.
(c) If the mass m is doubled and the force F remains the same, the acceleration a is reduced by one-half.

Such relation can be expressed by the following formula:
Acceleration a = net force / mass = F_net / m
Basically,

When F increases, a increases as well (if F doubles, a doubles)
When m increases, a decreases as well (if m doubles, a halves)

Example 1

Question: Calculate the acceleration of a 3000-kg, single-engine airplane as it begins its takeoff with an engine thrust of 1000 N.
Solution:
a = F_net / m = 1000 / 3000 = 0.33 m/s²

Exercise 1

Based on the given table; Find the corresponding acceleration.

Check your answers here: Solution

Example 2

Net force and the object's acceleration
If there is more than one vector acting on an object, the forces are added together as vectors,
taking into account their directions. See example below:
newton-second2

F_{to the right} = 9 N
F_{to the left} = 4 N
F_net = 9 N - 4 N = 5 N (to the right)
a = F_net / m = 5 N / 10 kg = 0.5 m/s² (to the right)

Forces acting in opposite directions

As per the below figure, let's calculate the mass m of the box if its acceleration a is 5 m/s²

newton-second3

Solution:
The net force is 30 N - 10 N = 20 N, directed to the right.
Using the formula F = m×a, the mass m = F/a = (20 N)/(5 m/s²) = 4 kg.

Exercise 2

Click here to access a similar exercise.
Check your answers here: Solution

Mass, Weight, and Inertia

A bigger force is required to produce the same acceleration for the bigger mass.
Inertia is known to be the object's resistance to a change in its motion.
Mass is a measure of an object's inertia.
The unit of mass is kilograms (kg).
An object's weight is the gravitational force acting on the object.
Weight is a force, measured in newtons (N).
In the absence of gravity, an object has no weight but having the same mass.

Example 3

Assume a ball hangs from a string attached to the ceiling as per the illustration below:
hanging-ball

Question: What would be the net force acting on the ball?

The net force is upward
The net force is zero
The net force is downward

Answer: The net force is zero.
Explanation: Since the ball is hanging from the ceiling at rest, it is not accelerating hence the net force becomes zero.
There are two following forces acting on the ball canceling each other: Tension from the string and force due to gravitation:

Frequently Asked Questions

Q: What is Newton's Second Law of Motion?

A: Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.
It is expressed by the formula: F = m × a, where F is force in Newtons, m is mass in kilograms, and a is acceleration in metres per second squared.

Q: What happens to acceleration if the force is doubled?

A: If the mass stays the same and the force is doubled, the acceleration doubles as well.
This is because acceleration is directly proportional to force according to Newton's Second Law.

What is the difference between mass and weight?

A: Mass is the measure of how much matter an object contains and is measured in kilograms (kg).
Weight is the gravitational force acting on that mass and is measured in Newtons (N). In the absence of gravity, an object would have no weight, but its mass remains the same.

What is inertia?

A: Inertia is an object's resistance to a change in its motion.
Mass is a measure of inertia — the greater the mass, the greater the force needed to accelerate it.

Q: What is net force?

A: Net force is the overall force acting on an object after all individual forces have been added together as vectors, taking direction into account.
If forces act in opposite directions, the smaller force is subtracted from the larger one to find the net force.

Q: What is 1 Newton equal to?

A: One Newton is defined as the force needed to accelerate a mass of 1 kilogram at a rate of 1 metre per second squared. In units: 1 N = 1 kg·m/s².

Introduction to Newton's Second Law