Unit 5: Torque and Rotational Motion

Torque:

Torque: A measure of the force that can cause an object to rotate about an axis

$$\tau =rFsin(\theta )$$

The Counter-clockwise torque is positive while clockwise torque is negative
Only the perpendicular component of force will apply a torque
The units of torque are Joules

Torque Applications:

Balance:

Remember to account for clockwise/counter-clockwise rotation when doing torque problems

To balance weights on a bar/seesaw, equal the sum of the torque on the left and the right must equal to zero
The closer the weight is to the center, the less torque it applies

$$\begin{gathered} \sum \tau =0 \\ {r}_a{F}_{a}sin(\theta )-r_b{F}_{b}sin(\theta )=0 \end{gathered}$$

Lever:

The closer the item being lifted is to the fulcrum, the less force is needed to move it

Screwdriver/Circle:

The greater the radius of a circle, the less force needed to exert the same amount of torque

Moment of Inertia:

Moment of Inertia: An object’s resistance to angular acceleration due to an applied torque

$$\begin{gathered} \sum {\tau }_i=I\alpha \\ \sum {\tau }_i=(\sum m_i{r}_i^2)\alpha \\ I=\sum m_i{r}_i^2 \end{gathered}$$

Angular acceleration and moment of inertia are inversly related
There are various values of I (Inertia) depending on the object’s shape and axis of rotation

Parallel Axis Theorem:

If the moment of inertia about the center of mass is known

$$I=I_{cm}+M{D}^2$$