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# Torque and Rotational Motion

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Torque is a word you may have heard but don’t fully understand. Well, what if I told you, even without understanding it, an average person demonstrates the concept of torque over 7,000 times a year by traveling to and from home, work, and any other activities throughout their day by the simple act of opening a door. This sounds crazy, right? Well, let’s take a dive deeper into this and learn how that’s possible.

Example of Torque: Door Hinge, adapted from public domain image.

This article will introduce the concepts of torque and rotational motion. Before going through some examples, we will define torque and rotational motion and how they are related to one another and angular momentum.

## What are the Definitions and Descriptions of Torque and Rotational Motion?

Below we will look at a couple of definitions and descriptions applicable to torque and rotational motion.

### Torque

The definition of torque is as follows:

The symbol $$\tau$$ represents torque, which is the amount of force applied to an object that will cause it to rotate about an axis.

The mathematical formula for torque is:

$$\tau = r F \sin{\theta}$$

where $$r$$ represents the radius measured in meters, $$\mathrm{m}$$, $$F$$ represents force measured in newtons, $$\mathrm{N}$$, and $$\theta$$ represents angular displacement. The SI unit for torque is newton meter, $$\mathrm{N}\,\mathrm{m}$$.

As torque is a vector, it has magnitude and direction, where its direction can either be clockwise or counterclockwise. The amount of torque applied to an object will always depend on two factors:

1. How much force is applied
2. The perpendicular distance from the axis of rotation

### Rotational Motion

In rotational motion, the components of velocity, acceleration, and displacement are the same form as their linear motion equivalents; however, we define them in terms of variables associated with rotational motion.

Rotational motion is a type of motion associated with objects traveling in a circular path. The force which causes these objects to travel in a circular path is known as torque.

We show the formulas for angular velocity, angular acceleration, and angular displacement below:

#### Angular velocity, $$\omega$$

$$\omega=\frac{\theta}{t}$$

where angular velocity is measured in radians per second, $$\frac{\mathrm{rad}}{\mathrm{s}}$$.

The derivative yields the equation $$\omega=\frac{d\theta}{dt}$$, which is the definition of instantaneous velocity.

#### Angular acceleration, $$\alpha$$

$$\alpha=\frac{\omega}{t}$$

where angular acceleration is measured in radians per second squared, $$\frac{\mathrm{rad}}{\mathrm{s}^2}$$.

The derivative yields the equation $$\alpha=\frac{d\omega}{dt}$$, which is the definition of instantaneous acceleration.

#### Angular displacement, $$\theta$$

$$\theta = \omega t$$

where we measure angular displacement in radians, $$\mathrm{rad}$$.

## Relationship between Torque and Rotational Motion and the formula for Rotational Torque

In linear motion, we know that objects move due to force. However, the force causing objects to move in rotational motion is known as torque. As a result, we can write the equation for torque in the same form as Newton’s second law, $$F=ma$$, and we show the formula for torque below:

$$\tau = I\alpha$$

where $$I$$ is the moment of inertia and $$\alpha$$ is angular acceleration.

## Can Angular Momentum be Related to Rotational Torque?

Besides solving problems with the equation for rotational torque, we can also use it to determine the relationship between variables simply by rearranging terms. For example, if asked what the relationship between angular acceleration and torque is, one can rearrange this equation to solve for angular acceleration and get the following:

$$\alpha = \frac{\tau}{I}$$

As a result, we can determine that angular acceleration is proportional to torque and inversely proportional to the moment of inertia. Now, we can go one step further by rewriting variables in terms of other variables. For example, if we know angular acceleration is equal to $$\alpha = \frac{\omega}{t}$$ and insert this into the equation for torque, we will get the following:

$$\tau = I \frac{\omega}{t} = \frac{I \omega}{t}$$

As a result, we can see another recognizable term associated with rotational motion. The term $$I\omega$$ represents angular momentum. Therefore, we can rewrite the equation for torque in terms of angular momentum as follows:

$$\tau = \frac{L}{t}$$

Therefore, without doing any mathematical calculations, we can determine the different relationships between variables associated with rotational motion.

### Torque Applied to Daily life

People demonstrate the concept of torque almost every day of their lives and may not even know it. Every time we open a door, we use the concept of torque as we cause the door to rotate on its hinges. From the formula for torque, as defined above, we know that torque is directly related to radius and force. Using this knowledge, we can understand why we place door handles at the farthest point from the door hinges. Let’s say that it requires $$100\,\mathrm{N}\,\mathrm{m}$$ of torque to open a door, and the distance from the hinges to the handle is $$2\,\mathrm{m}$$. We can then conclude that it would require $$50\,\mathrm{N}$$ of force to open the door. Now, if we move the door handle to the center of the door, the radius becomes $$1\,\mathrm{m}$$, and we would then have to apply a force of $$100\,\mathrm{N}$$ to open the door. This change demonstrates why radius is important to torque and why door handles are located at the farthest point from the hinges. Door handles at the farthest point allow for a maximum radius, allowing us to open doors with ease as we can apply less force. Door handles located in the center of a door would make opening doors harder for us because a smaller radius means we have to use more force for the door to open.

## Problems with Torque and Rotational Motion

To solve torque and rotational motion problems, the equation for torque can applied to different problems. As we have defined torque and discussed its relation to rotational motion, let us work through some examples to gain a better understanding of total mechanical energy. Note that before solving a problem, we must always remember these simple steps:

1. Read the problem and identify all variables given within the problem.
2. Determine what the problem is asking and what formulas apply.
3. Apply the necessary formulas to solve the problem.
4. Draw a picture if necessary to provide a visual aid

Using these steps, now let us work through some examples.

### Example 1

A plumber uses a $$2.5\,\mathrm{m}$$ wrench to loosen a bolt. If he applies $$65\,\mathrm{N}$$ of force, calculate how much torque is needed to loosen the bolt.

An example of a torque wrench. Adapted from Cdang, Wikimedia Commons, GFDL

After reading the problem, we are asked to calculate the torque needed to loosen a bolt and are given the radius of the wrench as well as the amount of force being applied. Therefore, using the formula for torque, our calculations are as follows:

\begin{aligned} \tau &= rF\sin{\theta} \\ \tau &= \left(2\,\mathrm{m}\right)\left(65\,\mathrm{N}\right)\sin{90} \\ \tau &= 130\,\mathrm{N}\,\mathrm{m} \end{aligned}

The amount of torque needed to cause the wrench to rotate and loosen the bolt is $$130\,\mathrm{N}\,\mathrm{m}$$.

Note that the plumber is applying a perpendicular force to the wrench, thus creating a $$90^{\circ}$$ angle.

### Example 2

An object, whose moment of inertia is $$45\,\mathrm{kg}/\mathrm{m}^2$$, rotates with an angular acceleration of $$3\,\mathrm{rad}/\mathrm{s}^2$$. Calculate the torque needed for this object to rotate about an axis.

An example of a rotating object with torque, StudySmarter Originals

After reading the problem, we are asked to calculate the torque needed for an object to rotate about an axis and are given the object's angular acceleration and moment of inertia. Therefore, using the formula for torque, our calculations are as follows:

\begin{aligned} \tau &= I\alpha \\ \tau &= \left( 45\,\mathrm{kg}/\mathrm{m}^2 \right) \left( 3\,\mathrm{rad}/\mathrm{s}^2 \right) \\ \tau &= 135\,\mathrm{N}\,\mathrm{m} \end{aligned}

The amount of torque needed to rotate the object about an axis is $$135\,\mathrm{N}\,\mathrm{m}$$.

## Torque and Rotational Motion - Key takeaways

• Torque is the force needed for an object to rotate about an axis.
• Rotational motion is the motion of objects traveling in a circular path.
• Rotational motion is associated with angular velocity, $$\omega$$, angular acceleration, $$\alpha$$, and angular displacement, $$\theta$$.
• We write the formula for torque in terms of radius and force, $$\tau = r F \sin{\theta}$$, and angular acceleration and moment of inertia, $$\tau = I\alpha$$.
• Torque occurs in our everyday life when we open doors.

The unit of torque in rotational motion is newton meter.

One example is that every time we open a door, we are using the concept of torque as we cause the door to rotate on its hinges

Torque is not equal to angular momentum, but it is related in that torque is angular momentum over time.

torque is related to angular acceleration by the formula torque equals moment of inertia times angular acceleration

torque equals force times distance

## Final Torque and Rotational Motion Quiz

Question

Even if the particles may have different translational acceleration, they can have the same angular acceleration.

True.

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Question

If the distance from the origin increases, what happens to the tangential acceleration assuming the angular acceleration stays constant?

It increases.

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Question

What is the right hand rule for the direction of torque?

Curl your hands around the axis of rotation with your fingers in the direction of the applied force. Your thumb then points in the direction of the torque. If your fingers curl counterclockwise torque is positive. If your fingers curl clockwise, torque is negative.

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Question

Define angular momentum and its units.

• Product of angular velocity and rotational inertia: $$L=I\omega$$.
• Units: $$\mathrm{\frac{kg\,m^2}{s}}$$.

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Question

Define torque and its units.

• Torque is the turning effect of a force: $$\tau = rF\sin\theta$$.
• Units: $$\mathrm{N\,m}$$.

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Question

Angular momentum is a conserved quantity. It is constant in ____

Closed systems.

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Question

The angular momentum of a system is constant over time if ____

the net external torque exerted on the system is zero.

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Question

What is rotational inertia?

Rotational inertia is an object's resistance to change angular velocity.

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Question

An object with high rotational inertia is _____ to rotate than one with a low one.

Harder.

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Question

What is a conserved quantity?

A quantity that's constant in closed/isolated systems.

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Question

Angular momentum is analogous to linear momentum

True, even though they have different units.

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Question

What is the relationship between net torque and angular momentum?

$$\tau_{\mathrm{net}}=\frac{\Delta L}{\Delta t}$$.

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Question

A frisbee spins without friction on someone's finger. An item is placed on top of the frisbee.

In which case will the angular velocity of the frisbee be smaller?

The item is placed on the outer edge.

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Question

Using conservation of angular momentum, explain this:

A star spins faster when it collapses because...

The moment of inertia of the star decreases as more mass is concentrated near its rotational axis. In turn, this increases its angular velocity.

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Question

A merry-go-round has a mass of 500 kg and a radius of 2.0 m. It rotates on a vertical axis through its center at 500 rev/min (52.4 rad/s). A child of 30 kg climbs on the outer edge of the merry-go-round. What is the angular velocity of the merry-go-round after the child climbs on it?

$$447\,\mathrm{\frac{rev}{min}}$$ $$\left(46.8\,\mathrm{\frac{rad}{s}}\right)$$

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Question

Which of the following is true about angular momentum?

We can assign angular momentum to a particle without defining a reference point.

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Question

Disk 1 rotates freely about an axis. We drop a non-rotating identical disk 2 on top. They stick together. What happens to the total angular momentum of the system?

Remains the same.

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Question

Suppose a spinning star collapses under its own gravitational force. The star's diameter is now 5 times smaller. What happens to its rotational speed?

Its rotational speed increases by a factor of 5.

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Question

Which of the following statements is not true about torque?

Torque is a rotational effect of a force and changes the rotational motion of an object about a fixed axis.

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Question

The magnitude of torque is written as...

$$\tau = r_{\perp}F = rF\sin\theta$$.

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Question

Torque has units of...

• $$\mathrm{N}\,\mathrm{m}$$
• $$\mathrm{\frac{kg\,m^2}{s^2}}$$.

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Question

In terms of rotational inertia, angular momentum is...

$$L=I\omega$$.

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Question

Angular momentum has units of

$$\mathrm{\frac{kg\,m^2}{s}}$$.

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Question

Which direction of the rotations is considered "positive" in Physics AP 1?

Counterclockwise

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Question

What is the relationship between the change of angular momentum and torque?

$$\Delta L = \tau \Delta t$$.

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Question

What is the relationship between net torque and angular acceleration?

$$\tau_{\mathrm{net}} = I\alpha$$.

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Question

What happens to an object's rotational inertia when we increase the distance between a mass and its axis of rotation?

The rotational inertia increases.

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Question

For a system to be balanced, the sum of clockwise torques must be...

Equal to the sum of counterclockwise torques.

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Question

Suppose an object has a moment of inertia $$I$$. If a net torque of $$\tau$$ is applied for 5 seconds, what is the object's angular velocity?

$$5I\tau$$.

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Question

The moment of inertia of a wheel is $$0.50\,\mathrm{kg\,m^2}$$. Find the angular acceleration in $$\mathrm{\frac{rad}{s^2}}$$ if its torque is $$1.5\,\mathrm{N\,m}$$.

$$3\,\mathrm{\frac{rad}{s^2}}$$

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Question

What is the definition of centripetal acceleration?

Centripetal Acceleration is the acceleration of an object undergoing rotational motion whose direction is always inward

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Question

What is the definition of centripetal force?

A centripetal force is a force applied to an object to keep it within a circular path, whose direction is always inward.

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Question

The word centripetal is defined by what phrase

Center-seeking

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Question

Centripetal acceleration is caused by a centripetal force.

True

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Question

What three external forces cause a centripetal force?

Friction

Tension

Gravity

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Question

The direction of centripetal acceleration is

inward, toward the center of the circle

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Question

The direction of a centripetal force is inward.

True

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Question

Centripetal acceleration and centripetal force are components of rotational motion.

True

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Question

What is the definition of torque?

Torque is defined as the force needed to cause an object to rotate about an axis.

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Question

Torque is a vector with magnitude and direction.

True.

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Question

Rotational motion describes the motion of objects traveling a linear path

False.

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Question

List three components associated with rotational motion.

angular velocity

angular acceleration

angular displacement

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Question

Angular acceleration is proportional to torque.

True.

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Question

What is one way people demonstrate the concept of torque every day?

Opening doors.

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Question

The amount of torque applied to an object, depends on what two factors:

1.  How much force is applied
2.  The perpendicular distance from the axis of rotation

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Question

What is angular motion?

The motion of a body along a curved path about a rigid point or axis.

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Question

True or false: angular motion includes both revolving and rotating motion.

True.

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Question

Which of the following formulas are true angular velocity equations?

$$\omega=\frac{\Delta \theta}{\Delta t}$$.

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Question

What does the variable angular displacement measure?

The separation angle between two points formed by the rotation of an object about a rigid axis.

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Question

What does the variable angular velocity measure?

The change in angular position over time.

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