2.2.3 Newton’s Second Law

• Newton's Second Law states:

The resultant force is equal to the rate of change in momentum. The change in momentum is in the same direction as the resultant force

• This can also be written as:

• This relationship means that objects will accelerate if there is a resultant force acting upon them
• This is derived from the definition of momentum as follows:

• An unbalanced force on a body means it experiences a resultant force
  • If the resultant force is along the direction of motion, it will speed up (accelerate) or slow down (decelerate) the body
  • If the resultant force is at an angle, it will change the direction of the body

Resultant Force

• Since force is a vector, every force on a body has a magnitude and direction
  • The resultant force is, therefore, the vector sum of all the forces acting on the body

• The direction is given by either the positive or negative direction as shown in the examples below

Resultant forces on a body can be positive or negative depending on their direction

• The resultant force could also be at an angle, in which case, the magnitude and direction of the resultant force can be determined using either:
  • Calculation (usually simple geometry, such as Pythagoras Theorem or the sine and cosine rules)
  • Scale drawing

Acceleration

• Since acceleration is a vector, it can be either positive or negative depending on the direction of the resultant force
  • If the resultant force is in the same direction as the motion of an object, the acceleration is positive
  • If the resultant force is in the opposite direction to the motion of an object, the acceleration is negative

• An object may continue in the same direction however with a resultant force in the opposite direction to its motion, it will slow down and eventually come to a stop
• If drag forces are ignored, or severely reduced, the acceleration is independent of the mass of an object
  • This has been shown in experiments by astronauts who have dropped a feather and a hammer on the Moon from the same height
  • Both the hammer and feather drop to the Moon's surface at the same time

Step 1: List the values given

• • Three possible forces at any angle of choice: 4 N, 8 N, and 24 N
  • Mass of object = 5 kg

Step 2: Consider the relevant equation

• • Newton's second law:

F = m × a

Step 3: Rearrange to make acceleration the focus

a = F ÷ m

Step 4: Investigate the minimum possible acceleration

• • It is best to consider the edges of this problem before dealing with more difficult combinations that is why it is prudent to check the minimum and maximum acceleration
  • The minimum would occur when the forces were acting against each other
  • The minimum possible would be when only the 4 N force was acting on the body
  • Now check the acceleration:

4 N ÷ 5 kg = 0.8 m s^–2

Step 4: Investigate the maximum possible acceleration

• • The maximum would occur when all three forces are acting in the same direction
  • Therefore adding to:

4 + 8 + 24 = 36 N

• • Now check the acceleration:

36 N ÷ 5 kg = 7.2 m s^–2

Step 5: Consider this range and the options

• • Since option D is higher than 7.2 m s^–2; it is not possible that these three forces can produce 10 m s^–2 acceleration for this mass
  • Option D is the correct answer