# Physics Study Guide/Torque

## Torque and Circular Motion

Circular motion is the motion of a particle at a set distance (called
radius) from a point. For circular motion, there needs to be a force
that makes the particle turn. This force is called the \'centripetal
force.\' Please note that the centripetal force is *not* a new type of
force-it is just a force causing rotational motion. To make this
clearer, let us study the following examples:

1.  If Stone ties a piece of thread to a small pebble and rotates it in
    a horizontal circle above his head, the circular motion of the
    pebble is caused by the tension force in the thread.
2.  In the case of the motion of the planets around the sun (which is
    roughly circular), the force is provided by the gravitational force
    exerted by the sun on the planets.

Thus, we see that the centripetal force acting on a body is always
provided by some other type of force \-- centripetal force, thus, is
simply a name to indicate the force that provides this circular motion.
This centripetal force is *always* acting inward toward the center. You
will know this if you swing an object in a circular motion. If you
notice carefully, you will see that you have to continuously pull
inward. We know that an opposite force should exist for this centripetal
force(by Newton\'s 3rd Law of Motion). This is the centrifugal force,
which exists only if we study the body from a non-inertial frame of
reference(an accelerating frame of reference, such as in circular
motion). This is a so-called \'pseudo-force\', which is used to make the
Newton\'s law applicable to the person who is inside a non-inertial
frame. e.g. If a driver suddenly turns the car to the left, you go
towards the right side of the car because of centrifugal force. The
centrifugal force is equal and opposite to the centripetal force. It is
caused due to inertia of a body.

$$\omega_{\text{avg}}=\frac{\omega+\omega_f}{2}=\frac{\theta}{t}$$

**Average angular velocity** is equal to one-half of the sum of
**initial** and **final angular velocities** assuming constant
acceleration, and is also equal to the **angle gone through** divided by
the **time taken**.

------------------------------------------------------------------------

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

**Angular acceleration** is equal to**change in angular velocity**
divided by **time taken**.

### Angular momentum

**Angular momentum** of an object revolving around an external axis $O$
is equal to the cross-product of the **position vector** with respect to
$O$ and its **linear momentum**.

**Angular momentum** of a rotating object is equal to the **moment of
inertia** times **angular velocity**.

$$L=I\omega$$

------------------------------------------------------------------------

$$\tau=I\alpha=\frac{\Delta L}{t}$$\>

**Rotational Kinetic Energy** is equal to one-half of the product of
**moment of inertia** and the **angular velocity** squared.

IT IS USEFUL TO NOTE THAT

The equations for rotational motion are analogous to those for linear
motion-just look at those listed above. When studying rotational
dynamics, remember:

-   the place of force is taken by torque

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-   the place of mass is taken by moment of inertia

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-   the place of displacement is taken by angle

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-   the place of linear velocity, momentum, acceleration, etc. is taken
    by their angular counterparts.

### Definition of terms

+----------------------------------------------------------------------+
| **Torque** ($\vec\tau$): Force times distance. A vector.             |
| $N\!\cdot\!m$                                                        |
|                                                                      |
| **Moment of inertia** ($I$): Describes the object\'s resistance to   |
| torque --- the rotational analog to inertial mass. $kg\!\cdot\!m^2$  |
|                                                                      |
| **Angular momentum** ($\vec L$): $kg\!\cdot\!\frac{m^2}{s}$          |
|                                                                      |
| **Angular velocity** ($\vec\omega$): $\frac{\text{rad}}{s}$          |
|                                                                      |
| **Angular acceleration** ($\vec\alpha$): $\frac{\text{rad}}{s^2}$    |
|                                                                      |
| **Rotational kinetic energy** ($K_r$):                               |
| $J=kg\!\cdot\!\left(\frac{m}{s}\right)^2$                            |
|                                                                      |
| **Time** ($t$): $s$                                                  |
+----------------------------------------------------------------------+
