# Newton's Second Law of Motion

## What Is Newton's Second Law?

**Newton's second law** of motion states the acceleration of an object is dependent on the net force acting on the object and the mass of the object.

**Mass**is defined as the quantity of matter in an object.

**Velocity**is defined as the speed of an object in a given direction. Two examples of velocity include a car traveling at 25 mph north or a plane flying at 575 mph in the northeast direction. An object that changes speed or direction has a change in velocity.- This rate at which velocity changes is defined as
**acceleration**. An object that changes direction or speed is accelerating. The change in speed can be increasing or decreasing to be considered acceleration. For example, a car that maintains a speed of 25mph but changes direction is considered accelerating. A car that maintains due north but either increases or decreases speed is considered accelerating. **Net force**is defined as the sum of all forces acting on an object. This image shows an object moving in the forward direction. {eq}F_{1} {/eq} and {eq}F_{2} {/eq} are vectors representing forces in two different directions. The net force (F) is the direction the object will move, taking account of all forces on the object.

### Newton's Laws of Motion

Isaac Newton was a physicist and mathematician who is best known for his three laws of motion.

#### Newton's First Law of Motion

Newton's first law of motion says an object at rest or an object in motion will remain at rest or in motion unless acted upon by an unbalanced force.

A **force** is a quantity with magnitude and direction. Forces are often represented by arrows. This image shows the two forces acting on the object. The size of the force is represented by arrow length. The net force on this object is zero because the length of arrows is equal in both directions representing a balanced force. According to Newton's first law, this object will remain at rest until the net force is no longer balanced. An unbalanced force is a force applied in opposite directions that are not equal in size or magnitude.

The following are true in order for an object to be considered at rest:

- The object has zero velocity.
- The forces acting on the object are balanced.
- The object has no acceleration.

The first law of motion is also referred to as the law of inertia. **Inertia** describes the

resistance an object has to a change in the state of motion. For example, an elephant and rabbit are running at the same speed. The elephant will have greater inertia than the rabbit because the force needed to change the motion of the elephant will be greater than the force needed to change the motion of the rabbit.

#### Newton's Second Law of Motion

Newton's second law of motion states that the acceleration of an object is dependent on the net force acting on the object and the mass of the object. This law is represented by the equation: {eq}F_{net} {/eq}=ma. The equation states that the net force of an object is determined by the product of its mass and acceleration.

#### Newton's Third Law of Motion

Newton's third law of motion states that for every action, there is an equal and opposite reaction. Forces come in pairs. The force this swimmer applies to the water is paired with an equal and opposite force applied by the water. The force of the water in response to the force applied by the swimmer's arms propels the swimmer forward.

Newton's second law states that the net force of an object is the product of its mass (m) and acceleration (a). This force equation relates: force mass x acceleration:

{eq}F_{net} {/eq}=ma

The International System of Units is used to describe the units for each variable in Newton&pos;s second law: Net force ({eq}F_{net} {/eq}) is measured in Newtons (N). The units of Newtons are {eq}kg \,m/s^{2} {/eq}. Net force is the vector sum of all the forces acting on the object. Acceleration (a) is measured in ({eq}m/s^{2} {/eq}). Mass (m) is measured in kilograms (kg).

In order to find the vector sum, the sum of all the forces is calculated.

**Example 1:**

An object being pushed to the left with a force of 15N and pushed to the right with a force of 10N would have a net force of 5N to the left.

**Example 2:**

This image shows an object with force A and force B pulling in opposite directions. If these forces had quantitative values, the net force (C) would be in the direction of force A. This is because the magnitude of force A is larger than force B.

The equation {eq}F_{net}=ma {/eq} shows the relationship between net force, mass, and acceleration. As mass increases, the net force increases. As the acceleration of an object increases, the net force on the object also increases. If either mass or acceleration decreases, the net force will also decrease.

### Calculating for Force

This example shows how to use Newton's second law to calculate force.

## Newton's Second Law of Motion

Consider two balls, one with a mass of 1 kg and the other with a mass of 10 kg. Which ball would experience a greater change in motion if kicked with the same force? Clearly, the smaller ball would experience a greater change in motion. An object's state of motion can be described as its velocity, where **velocity** is the speed of an object with respect to its direction. Objects at rest - for example, the balls you see on the screen - have zero velocity. Once kicked, the ball's state of motion changes. In other words, its velocity changes. *When an object changes its velocity*, it has what we call **acceleration**. **Newton's second law of motion** provides the explanation for the behavior of objects when forces are applied. The law states that external forces cause objects to accelerate, and the amount of acceleration is directly proportional to the net force acting on the objects and inversely proportional to the mass of the objects.

An object with a mass of 2.5 kg is accelerating at 3.0 {eq}m/s^{2} {/eq}. What is the net force?

Using the equation: {eq}F_{net} {/eq}=ma, plug in the known values:

{eq}F_{net} {/eq}= (2.5kg)({eq}3.0 m/s^{2} {/eq})

{eq}F_{net} {/eq}= 7.5N

### Calculate for Acceleration

Rearranging Newton's second law of motion, we can see that acceleration is directly proportional to the net force acting on an object and inversely proportional to the mass of the object. {eq}a = F_{net}/m {/eq}

This example shows how to use Newton's second law to calculate acceleration.

An object with a mass of 3.5 kg hits a wall with a force of 2.0N. What is the acceleration of the object?

Rearrange the equation: {eq}F_{net} = ma {/eq} to solve for (a)

This is done by dividing both sides by mass (m)

{eq}F_{net}/m= a {/eq}

The equation can now be used to calculate acceleration: {eq}a = F_{net}/m {/eq}

Plug in the known values:

a=(2.0N)/(3.5kg)

a=0.57{eq}m/s^{2} {/eq}

## Newton's Second Law Examples

**Example 1:** Calculating net force

An object with a mass of 3.5 kg is accelerating at 2.5{eq}m/s^{2} {/eq} . What is the net force?

Using the equation: {eq}F_{net} {/eq}=ma, plug in the known values:

{eq}F_{net} {/eq}= (3.5kg)({eq}2.5 m/s^{2} {/eq})

{eq}F_{net} {/eq}= 8.8N

**Example 2:** Calculating acceleration

An object with a mass of 1.5 kg hits a wall with a force of 3.2N. What is the acceleration of the object?

Rearrange the equation {eq}F_{net} = ma {/eq} to solve for (a)

This is done by dividing both sides by mass (m)

{eq}F_{net}/m = a {/eq}

The equation can now be used to calculate acceleration: {eq}a = F_{net}/m {/eq}

Plug in the known values:

a=(3.2N)/(1.5kg)

a=2.1{eq}m/s^{2} {/eq}

**Example 3:** Calculating mass

An object accelerating at 4.0{eq}m/s^{2} {/eq} hits a wall with a force of 3.5N. What was the mass of the object?

Rearrange the equation {eq}F_{net} = ma {/eq} to solve for (m)

This is done by dividing both sides by mass (a)

{eq}F_{net}/a = m {/eq}

The equation can now be used to calculate mass: {eq}m = F_{net}/a {/eq}

Plug in the known values:

{eq}m = (3.5N)/4.0m/s^{2} {/eq})

m=0.88 kg

## Importance of Newton's Second Law of Motion

Newton's second law of motion provided the foundation for scientific inquiry. His work on motion has been coined as one of the most important works in modern science. In 1687, Newton published his laws of motion, which greatly influenced the Enlightenment period in Europe. Up until the 20th century, Newton's laws were regarded as the most fundamental laws of physics. As the understanding of physics evolved, quantum mechanics and relativity became the most fundamental base of physics.

## Lesson Summary

**Newton's second law** of motion states the acceleration of an object is dependent on the net force acting on the object and the mass of the object. **Mass** is defined as the quantity of matter in an object and is measured in kilograms (kg). **Velocity** is defined as the speed of an object in a given direction and is measured in m/s. The rate at which velocity changes is defined as **acceleration** and is measured in ({eq}m/s^{2} {/eq}).

Newton's second law states net force is equal to the product of an object's mass and acceleration: {eq}F_{net} = ma {/eq}. **Net force** is defined as the sum of all forces acting on an object and is measured in Newtons (N) or ({eq}kg \,m/s^{2} {/eq}). A **force** is a quantity with magnitude and direction and is represented with vectors. The net force is calculated by subtracting forces in opposite directions. An object is considered at rest if any of the following occur: the object has zero velocity, the forces acting on the object are balanced, or the object has no acceleration.

## What is Net Force?

Notice that **Newton's second law** states that the amount of acceleration is directly proportional to the net force acting on the object. What is the net force and how do we calculate it? **Net force** is the sum of all forces acting on an object in a particular direction. Since forces have direction, they are vector quantities. **Vector quantities** are fully described with both magnitude and direction and are represented with arrows. Consider an object being pushed to the left with 10 Newtons of force and to the right with 5 Newtons of force. The **net force = 5 N to the left** as 10 N - 5 N = 5 N. The forces are subtracted from each other since they are pointing in opposite directions. The forces would be added to each other if they were pointing in the same direction. Once the net force is determined, the acceleration of the object can be determined.

## How to Calculate for Acceleration

**Acceleration** is a *change in velocity*. As long as we know the mass of the object and the net force acting on the object, we can determine acceleration. Let's look at the formula:

**a = f (net) / m**, where *a = acceleration, f (net) = the net force acting on the object, m = the mass of the object*.

#### Acceleration Example

Let's look at an example. Consider a 10 kg object forced to the left with 10 Newtons and to the right with 20 Newtons. What is the acceleration of this object?

Let's recall the formula for acceleration. **a = f (net) / m**

Let's first calculate the net force. As the forces are acting in opposite directions, we subtract them.

**f (net) = 20 N right - 10 N left**, and we get**f (net) = 10 N right**

Now, plug net force into the equation with mass to calculate for acceleration.

**a = 10 N / 10 kg** or **a = 1 N/Kg **

A Newton of force equals 1 kg * m/sec^2. **1 N = 1 kg * m/sec^2**. If we substitute this value of a Newton into the equation, mass (kg) is canceled out, and that leaves us with the units **m/sec^2**. Those are the *units for acceleration*.

Therefore the acceleration of our object is, **a = 1 m/sec^2**.

## How to Calculate for Force

If we know the mass and acceleration of an object, we can calculate for force. Simply rearrange the equation to solve for force. So, as you see on the screen, **f (net) = m * a**.

#### Force Example

Consider again our **10 kg object**, now moving to the west with an acceleration of **10 m/sec^2**. What is the net force acting on this object?

Recall the equation for force. **f (net) = m * a**. So we simply multipy 10 kg, the mass of our object, by 10 m/sec^2, the acceleration of our object. **f (net) = 10 kg * 10 m/sec^2**.

And that gives us a net force of 100 kg * m/sec^2. **f (net) = 100 kg * m/sec^2**. If we recall those units, kg * m/sec^2, are equal to a Newton so the net force equals 100 Newtons. **f (net) = 100 N**.

Therefore, our object is pushed to the west with a net force of 100 N.

## Implications of Newton's Second Law of Motion

Newton's second law implies that acceleration occurs only in the presence of an unbalanced force. If the forces acting on an object are balanced, the object is in a constant state of motion. That is, it is not accelerating. This is true for objects at rest as well as objects moving with a constant velocity. For example, a parked car has a net force of zero. The forces are balanced as the force of gravity pulling down is canceled out by the force of the ground pushing up on the parked car. Likewise, a car moving with a constant velocity of 65 MPH is experiencing balanced forces as well. If an unbalanced force is applied, the car will accelerate. If you apply the brakes, the car will slow down - that is, experience negative acceleration. If you apply the gas, the car will increase its velocity - that is, experience positive acceleration.

## Lesson Summary

**Newton's second law of motion** provides an explanation for the behavior of objects when forces are applied to the objects. The law states that *external forces cause objects to accelerate, and the amount of acceleration is directly proportional to the net force and inversely proportional to the mass of the object*. **Net force** is the *sum total of all forces acting on an object in a particular direction*. Forces acting in the same direction can be added together while forces acting in opposite directions are subtracted from each other to determine the net force. **Acceleration** is a *change in velocity*. The formula for calculating acceleration is as follows: **a = f (net) / m**, where *a = acceleration, f (net) = the net force acting on the object, m = the mass of the object*. Force can be calculated by simply rearranging the formula to solve for force, as you can see on the screen, **f (net) = m * a**. If all the forces acting on an object are balanced - that is, the net force is zero - the object does not accelerate. If an unbalanced force is applied, then the object will accelerate.

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## Newton's Second Law of Motion

Consider two balls, one with a mass of 1 kg and the other with a mass of 10 kg. Which ball would experience a greater change in motion if kicked with the same force? Clearly, the smaller ball would experience a greater change in motion. An object's state of motion can be described as its velocity, where **velocity** is the speed of an object with respect to its direction. Objects at rest - for example, the balls you see on the screen - have zero velocity. Once kicked, the ball's state of motion changes. In other words, its velocity changes. *When an object changes its velocity*, it has what we call **acceleration**. **Newton's second law of motion** provides the explanation for the behavior of objects when forces are applied. The law states that external forces cause objects to accelerate, and the amount of acceleration is directly proportional to the net force acting on the objects and inversely proportional to the mass of the objects.

## What is Net Force?

Notice that **Newton's second law** states that the amount of acceleration is directly proportional to the net force acting on the object. What is the net force and how do we calculate it? **Net force** is the sum of all forces acting on an object in a particular direction. Since forces have direction, they are vector quantities. **Vector quantities** are fully described with both magnitude and direction and are represented with arrows. Consider an object being pushed to the left with 10 Newtons of force and to the right with 5 Newtons of force. The **net force = 5 N to the left** as 10 N - 5 N = 5 N. The forces are subtracted from each other since they are pointing in opposite directions. The forces would be added to each other if they were pointing in the same direction. Once the net force is determined, the acceleration of the object can be determined.

## How to Calculate for Acceleration

**Acceleration** is a *change in velocity*. As long as we know the mass of the object and the net force acting on the object, we can determine acceleration. Let's look at the formula:

**a = f (net) / m**, where *a = acceleration, f (net) = the net force acting on the object, m = the mass of the object*.

#### Acceleration Example

Let's look at an example. Consider a 10 kg object forced to the left with 10 Newtons and to the right with 20 Newtons. What is the acceleration of this object?

Let's recall the formula for acceleration. **a = f (net) / m**

Let's first calculate the net force. As the forces are acting in opposite directions, we subtract them.

**f (net) = 20 N right - 10 N left**, and we get**f (net) = 10 N right**

Now, plug net force into the equation with mass to calculate for acceleration.

**a = 10 N / 10 kg** or **a = 1 N/Kg **

A Newton of force equals 1 kg * m/sec^2. **1 N = 1 kg * m/sec^2**. If we substitute this value of a Newton into the equation, mass (kg) is canceled out, and that leaves us with the units **m/sec^2**. Those are the *units for acceleration*.

Therefore the acceleration of our object is, **a = 1 m/sec^2**.

## How to Calculate for Force

If we know the mass and acceleration of an object, we can calculate for force. Simply rearrange the equation to solve for force. So, as you see on the screen, **f (net) = m * a**.

#### Force Example

Consider again our **10 kg object**, now moving to the west with an acceleration of **10 m/sec^2**. What is the net force acting on this object?

Recall the equation for force. **f (net) = m * a**. So we simply multipy 10 kg, the mass of our object, by 10 m/sec^2, the acceleration of our object. **f (net) = 10 kg * 10 m/sec^2**.

And that gives us a net force of 100 kg * m/sec^2. **f (net) = 100 kg * m/sec^2**. If we recall those units, kg * m/sec^2, are equal to a Newton so the net force equals 100 Newtons. **f (net) = 100 N**.

Therefore, our object is pushed to the west with a net force of 100 N.

## Implications of Newton's Second Law of Motion

Newton's second law implies that acceleration occurs only in the presence of an unbalanced force. If the forces acting on an object are balanced, the object is in a constant state of motion. That is, it is not accelerating. This is true for objects at rest as well as objects moving with a constant velocity. For example, a parked car has a net force of zero. The forces are balanced as the force of gravity pulling down is canceled out by the force of the ground pushing up on the parked car. Likewise, a car moving with a constant velocity of 65 MPH is experiencing balanced forces as well. If an unbalanced force is applied, the car will accelerate. If you apply the brakes, the car will slow down - that is, experience negative acceleration. If you apply the gas, the car will increase its velocity - that is, experience positive acceleration.

## Lesson Summary

**Newton's second law of motion** provides an explanation for the behavior of objects when forces are applied to the objects. The law states that *external forces cause objects to accelerate, and the amount of acceleration is directly proportional to the net force and inversely proportional to the mass of the object*. **Net force** is the *sum total of all forces acting on an object in a particular direction*. Forces acting in the same direction can be added together while forces acting in opposite directions are subtracted from each other to determine the net force. **Acceleration** is a *change in velocity*. The formula for calculating acceleration is as follows: **a = f (net) / m**, where *a = acceleration, f (net) = the net force acting on the object, m = the mass of the object*. Force can be calculated by simply rearranging the formula to solve for force, as you can see on the screen, **f (net) = m * a**. If all the forces acting on an object are balanced - that is, the net force is zero - the object does not accelerate. If an unbalanced force is applied, then the object will accelerate.

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#### What is Newton's second law of motion in simple terms?

Newton's second law of motion states the acceleration of an object is dependent on the net force acting on the object and the mass of the object. This law is represented by the equation: Fnet=ma.

#### What are Newton's 1st, 2nd, and 3rd laws?

Newton's first law of motion says an object at rest or an object in motion will remain at rest or in motion unless acted upon by an unbalanced force. Newton's second law of motion states the acceleration of an object is dependent on the net force acting on the object and the mass of the object. Newton's third law of motion states for every action, there is an equal and opposite reaction.

#### What is an example of Newton's second law?

An object with a mass of 2.5 kg is accelerating at 3.0 m/s^{2} . What is the net force?

Fnet= (2.5kg)(3.0 m/s^{2})

Fnet= 7.5N

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