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.

${\displaystyle \omega _{\text{avg}}={\frac {\omega +\omega _{f}}{2}}={\frac {\theta }{t}}}$

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.

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

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

Angular momentum

 ${\displaystyle {\vec {L}}={\vec {r}}\times {\vec {p}}=m({\vec {r}}\times {\vec {v}})}$

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

 ${\displaystyle {\vec {L}}=I{\vec {\omega }}}$

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

${\displaystyle L=I\omega }$

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

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
• the place of mass is taken by moment of inertia
• the place of displacement is taken by angle
• the place of linear velocity, momentum, acceleration, etc. is taken by their angular counterparts.

Definition of terms

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