Newton's laws

In a nutshell

Newton's laws of motion are three very important laws that describe how objects move. They relate the forces that act on an object to the resultant motion of the object. The impulse is the change in momentum of an object.

Equations

Variable definitions

Newton's first law

Newton's first law states that 'every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force.'

A change in its motion could be the object speeding up or slowing down, it could also be the object starting to move from rest. It could also be a change in direction.

If the resultant force on an object is zero, then it will remain in its current state of motion. If it was stationary, then it will remain stationary after the resultant force is applied.  If it was moving, then it will continue to move at the same velocity after the resultant force is applied. 

Newton's second law

Newtons second law states that the force is proportional to the rate of change of momentum. 

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It is given by:

​$$F = \dfrac{\Delta P}{\Delta t}$$

A special case of Newton's second law, when the mass of an object is constant is:

​$$F = \dfrac{\Delta P}{\Delta t}\newline \\[0.15in] F= \dfrac{m\Delta v}{\Delta t}\newline \\[0.15in] a = \dfrac{v}{\Delta t}\newline \\[0.15in] F =ma$$

Note: $$F=ma$$ cannot be used for objects with changing mass, such as rockets.​

Example

A ball has a mass of $$400 \space g$$​. If the ball experiences a resultant force of $$10 \space N$$​, what is the acceleration it will experience? 

Write out the quantities you have been given and make sure they are in the correct form: 

​$$m = 400 \space g = 0.4 \space kg$$

$$F = 10 \space N$$

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Then write down the equation you need to use: 

​$$F = m a$$

Rearrange the equation to solve for acceleration instead: 

​$$a = \dfrac{F}{m}$$

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Finally, substitute the values into the equation: 

​$$a = \frac{10}{0.4}$$

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Therefore, the acceleration of the ball is $$\underline{ 25 \ ms^{-2}}$$​.

Newton's third law

The third law states that when two objects interact with one another, the forces that they exert on each other are equal and opposite. Another way to think about this law is that, for every action, there is an equal and opposite reaction.

Newton's third law must be the same type of force acting on two different objects. 

Example

If a person pushes against a wall, they exert a force onto the wall. As a result, they will experience a reaction force from the wall. This reaction force will push against their hands. It will be a reaction force of the same magnitude as the action, but in the opposite direction.

1. Reaction force 2. Action force

Note: The forces are the same type but acting on different objects. One is the force of the person on the wall and the other is the wall on the person. This is also an example of Newton's first law as the forces are balanced, which means the person is stationary as they weren't moving beforehand. 

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Impulse

An impulse is the change in momentum of an object. By rearranging Newton's 2nd Law, you can find an equation for the impulse on an object:

​$$F= \dfrac{(\Delta p)}{\Delta t}\newline \\[0.1in] F = \dfrac{(\Delta mv)}{\Delta t}\newline \\[0.1in] F {\Delta t} = (\Delta mv)$$​​

Where the impulse is $$F \Delta t$$. This is also equal to the area of a force-time graph, therefore the area of a force-time graph equals the change in momentum:

1. Force 2. Time 3.  Change in momentum

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Example

A ball is hit with a cricket bat with a force of 120 N and there is an impact time of 0.3 s. What is the change in momentum of the ball?

First, state the variables from the question:

$$F=120 \ N, \ \ t=0.3 \ s$$

Then state the equation needed:

$$F {\Delta t} = (\Delta mv)$$​​

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Finally, substitute into the equation and solve:

$$(\Delta mv) = 120 \times 0.3\newline \\[0.1in] = 36 \ kgms^{-2}$$

The change in momentum of the ball is $$\underline{36 \ kgms^{-2}}$$​​