Forces and acceleration

In a nutshell

According to Newton's second law, force and acceleration are directly proportional for an object with a constant mass. Resultant forces change an objects acceleration. Gravity gives objects with mass weight.

Equations

Variable Definitions

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Newton's second law

Newton's second law is as follows:

The acceleration of an object is proportional to the resultant force on the object.

Therefore, to change the motion of an object, or to accelerate that object, a force must be applied. The force applied is proportional to the mass of the object being acted on multiplied by the acceleration produced. Newton's second law is:

​$$\boxed{F=ma}$$

An object with twice the mass of another object will require twice the force to accelerate it to the same extent.

Weight

The force of Earth's gravity acts on all objects on or around the Earth's surface. The acceleration due to this gravitational force is known as $$g$$, and on Earth it's value is $$9.81\ ms^{-2}$$. Since that acceleration is constant, objects of different mass will have a different magnitude of force acting on them to accelerate them by $$9.8\space ms^{-2}$$ towards Earth. This force is known as weight:

​$$\boxed{W=mg}$$

An object of greater mass does not fall faster than a lighter one, but the weight force acting on them is different.

Example 1

An object of mass $$m\space kg$$ is falling towards the Earth with an acceleration of $$3 \ ms^{-2}$$​.  A force of $$50N$$ is acting against its weight force. What is the value of $$m$$? Give you answer in $$3\space s.f.$$

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Identify the forces acting on the object. Take down to be positive.

$$\begin{aligned}&F\uparrow=50\space N\\&F\downarrow =mg\ N\\&F= (mg-50)\end{aligned}$$

Use $$F=ma$$ to solve for $$m$$​

$$\begin{aligned}F&=ma\\a&=3\ ms^{-2}\\g&=9.8\ ms^{-2}\\F&=(mg-50)\\9.8m-50 &=3m\\6.8m&=50\\m&=\dfrac{50}{6.8}\\&\underline{m=7.35 \space kg} \ (3 \ s.f.)\end{aligned}$$