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CIE A Level Maths: Mechanics复习笔记3.2.1 F = ma

Category: A-level课程, 教材笔记, 福利干货 Date: 2022年8月17日下午5:20

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 3.1.1 Equilibrium in 1D and 3.1.2 Equilibrium in 2D but has been included here for completeness

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.

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”

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.

In most problems where forces are acting in different directions, those directions will be perpendicular to one another or can be resolved into components acting perpendicular to each other
In such cases we apply N2L (F = ma) and ‘suvat’ equations separately to both directions

s2kauWJB_3-2-1-fig1-forces-h-and-v

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
- 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 = -10 m s^-2

3.2.1_WE_RN_F-Ma_1

(a) Find the acceleration of the train engine.

3-2-1-fig3-we-solution-a

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

3-2-1-fig3-we-solution-b

3-2-1-fig3-we-solution-c

Sketching, or adding to given, diagrams can help to understand problems and can help you decide which direction to take as positive.
Remember that F (in N2L) is the resultant force, sometimes seen as F_net- be careful not to get it muddled with any other forces that are, or could be, denoted by F. To avoid confusion, use quote marks around “F = ma" to show that the quoted F, m and a do not necessarily correspond to F, m or a mentioned in the question.
Depending on which direction is taken as positive, the resultant force, F N , may be negative and/or acceleration, a m s^-2, may be negative (this is particularly relevant for vertical motion)