F = ma (AQA A Level Maths: Mechanics)

What is Newton’s First Law of Motion (N1L)?

• An object at rest will stay at rest, and an object moving with constant velocity will continue to move with constant velocity, unless an unbalanced force acts on the object
• This law is explored more in 1.1 Equilibrium in 1D and 3.1.2 Equilibrium in 2D but has been included here for completeness

What is Newton’s Second Law of Motion (N2L)?

• The resultant force (F_net) acting on a body is equal to the product of the mass of the body and its acceleration
  • F = ma 
    • F is the resultant force (N)
    • m is the mass (kg)
    • a is the acceleration (m s^-2)
• This will probably be the most familiar of Newton’s Laws of Motion as it has an equation (F = ma) that you will use frequently in mechanics problems.

What is Newton’s Third Law of Motion (N3L)?

• For two bodies, the force exerted on the second by the first is equal in magnitude but opposite in direction to the force exerted on the first body by the second
• This is sometimes loosely referred to as “for every action there is an opposite and equal reaction”

When do I use F = ma (Newton’s Second Law or N2L)?

• Use it to set up and solve equations when motion is involved
• Some related equations may come from the constant acceleration equations (‘suvat’) but F= ma is needed when force(s) and mass are mentioned or involved (neither force nor mass are involved in the ‘suvat’ equations)
• If not asked directly in a question it will be implied by the information given – motion and acceleration will be involved and the mass of the particle will be relevant too

How do I solve problems using F = ma(N2L) and the ‘suvat’ equations?

• F = ma can be used in conjunction with the ‘suvat’ equations – the linking connection is acceleration (a)
• ‘suvat’ only questions will not involve mass or (resultant) force
• Step 1. Draw a diagram and label all forces acting on the particle(s)
  • label the positive direction and any other useful information
  • If a diagram is given, add anything missing to it
• Step 2. Use N2L, F = ma , or an appropriate ‘suvat’ equation.
  • If there is more than one particle involved you may have to do this for each
• Step 3. Solve the equation
  • In harder problems simultaneous equations may arise.

How do we deal with forces acting in different directions on a particle in motion?

• At AS level, if forces are acting in different directions, those directions will be perpendicular to one another
  • Thus nearly all questions at AS level involve forces acting horizontally (x - direction) and/or vertically (y -direction)
• In such cases we apply N2L (F = ma) and ‘suvat’ equations separately to both directions

How do we use F = ma in problems involving weight?

• Weight is a force
  • W = mg N where g m s^-2 is the acceleration due to gravity
    • If you are told, or are using g = 9.8 m s^-2 ensure you round your final answer to two significant figures (as 9.8 is rounded to 2 s. f.)
    • If you are told to use g = 10 m s^-2 ensure you round your final answer to one significant figure (as 10 is rounded to 1 s. f.)
  • Weight always acts vertically downwards (towards earth)
  • If upwards is the positive direction (and assuming no other vertical forces are involved) then acceleration would be negative, a = -9.8 m s^-2

(a)  Find the acceleration of the train engine.

(b)  Find the resultant force acting on the train engine.

(c)  Given that the (only) driving force is 6250 N, find the total of any resistive forces acting on the train engine.