 Newton's Laws of Motion # Newton's Second Law of Motion

In the material, Uniformly accelerated motion, we learned that the speed of a body changes with acceleration or deceleration. But what is the cause of acceleration and deceleration?

What causes the change in the speed or direction of a body is the force acting on it.

Example

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In the above example, the soccer player, with the force of his legs, acted on the soccer ball. Let's imagine replacing the soccer ball with a heavy medicine ball. In this case, the soccer player would have to use a greater force on the ball to set it in motion.

In the following, we will learn how the physical quantities listed below are related to each other:

• the mass of the body,

• the force acting on the body, and

• the acceleration of the body.

Here we will consider the motion where the force acts in the same or in the opposite direction to the motion of the body.

## Force, mass, and acceleration

Let's take a trolley with mass , which can move without friction on a horizontal surface, as shown in the figure below. We stand against the trolley and push it in front of us with a constant force . The trolley moves uniformly accelerated with acceleration .

Now, we load the trolley with a load so that the total mass of the trolley and the load is twice as large, i.e. . By measuring the force, we find out that we have to push the trolley with twice the force, i.e. with a force so that the trolley moves with the same acceleration.

With twice the mass, we need twice the force for the same acceleration. Let's state the result in another way. The acceleration of a body is:

Let's write down the conclusion in mathematical form: Let's check the accuracy of the above equation. Let's take twice the mass and denote it as and twice the force denoted as : We note that:   We cancel out 2 on the right side of the equation. We see that the acceleration of the body does not change if we simultaneously double the:

• mass and

• force.

Let's rearrange the formula for the acceleration by making the force the subject. We multiply both sides of the equation by the mass : We cancel out on the right-hand side of the equation: We switch both sides of the equation: The formula: represents Newton's second law. If we translate it into words, it can be understood as follows: A body with mass moves uniformly accelerated with acceleration if the resultant force act on it.

Using the above equation, we can express the unit for force using basic units of measurement. We do this by inserting units for individual quantities into the equation: The resultant of the forces can be:

• positive

A positive resultant means that it acts in the same direction as the motion of the body. In this case, the body moves with uniform acceleration (acceleration is positive).

• negative

A negative resultant means the action of the force is in the opposite direction to the motion of the body. In this case, the acceleration is also negative. The motion is steadily decreasing.

Let the resultant of all the forces acting on a body with mass be equal to . Then the body moves uniformly accelerated with acceleration . This is mathematically expressed as: A positive force acts in the direction of motion and accelerates the body while a negative force acts in the opposite direction to the motion and slows the body down.

### A force acting in the direction of motion

Let's look at some examples where a force acts in the direction of the motion of a body. The force is positive and accelerates the body.

Example

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Example

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Let's look at the example where we calculate the acceleration of a car.

Example

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### A force acting opposite to the direction of motion

If a force acts in the opposite direction to the motion of a body, the body moves uniformly decelerating. Let's look at a few examples of decelerating motion.

Example

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Example

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Let's look at an example in which we calculate the braking force.

Example

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## Weight

Gravity is the force with which the Earth attracts all objects. The phenomenon is called weight or gravity.

Example

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Gravity acts not only on Earth but also on other celestial bodies. The Sun, the planets, and their moons have a gravity that is different from that on Earth.

Example

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The force of gravity will be denoted by . The subscript stands for gravity.

In the material, Free fall, we saw that a body moves with uniform acceleration during free fall. The acceleration of a body in free fall is called free fall acceleration or gravitational acceleration and we gave it the symbol . It is approximately: According to Newton's second law, weight can also cause a body to move.

Let's write Newton's second law: We insert gravitational force for force and gravitational acceleration for acceleration :   We have obtained a version of Newton's second law for the weight of bodies. Weight is therefore equal to the product of mass and acceleration of free fall. Since the acceleration of free fall is approximately , we can say that the weight in Newtons is numerically ten times the mass in .

Weight or gravitational force is the force with which the Earth (or another celestial body) attracts all bodies. We calculate it as the product of mass and gravitational acceleration : Example

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Let's look at how weight affects bodies:

• at rest,

• in free fall,

• on a slope.

We will see that the motion of a body depends only on the base due to its weight.

### A body resting on a horizontal surface

For a body to be at rest, the resultant of all forces acting on the body must be equal to zero. If the weight acts vertically downward, there must also be an equally large force acting vertically upward. This is the force with which the surface acts on the body. We call it the normal force and is denoted by .

Example

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### A body falling freely

If the normal force is less than the weight of the body, the body moves in the direction of their resultant, i.e. vertically downwards.

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However, if the force of the ground is zero (no ground or base), the body falls freely. Its motion is uniformly accelerated with an acceleration .

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### A body moving down a slope

Let's place a body on a slope and see what kind of motion is caused by the weight of the body alone. The weight acts vertically downwards, towards the centre of the Earth. Let's resolve it into two perpendicular components:

• the static component, and

• the dynamic component.

The static component of the weight acts perpendicular to the slope and presses the body against the slope. The slope, on the other hand, acts on the body with an opposite equal force . The forces and are in equilibrium (they cancel each other out), so the body does not move in a direction perpendicular to the slope.

The dynamic component of the weight acts along the slope. If no other external forces act on the body (friction, resistance, pulling, and pushing forces), then is the resultant of all the forces acting on the body. Therefore, according to Newton's second law, it causes the body to move with uniform acceleration down the slope. The steeper the slope, the greater the dynamic component and the greater the acceleration.

Example

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The slope can be imagined as a gentle free fall. In the extreme case, if the slope is vertical, the motion turns into a free fall.