# Torque

Helena Dedic

If you need to understand the concept of center of mass, please see the link below:
CENTER OF MASS

• A force causes a free object to move in a translational motion if it is applied at the centre of mass of the object or the line of force passes through the centre of mass.

• When a force is not applied at the centre of mass of a free object or when the line of force does not pass through the centre of mass then it causes this object to move in a complex way that involves both translation (i.e.: movement of the center of mass) as well as rotation (i.e.: movement around the center of mass.)

• OBSERVATIONS OF AN OBJECT:
a) When a force acts at the pivot or when a line of force passes through the pivot,
the object does not begin to rotate.


b) When a force acts at some distance from the pivot, the object begins to rotate.


• Lever arm is a vector that points from the pivot to the point where the force acts.

• TORQUE: A cause of a change of the angular velocity (e.g.: a rotating wheel stops or it begins to rotate faster.
The symbol for torque is ${\displaystyle {\vec {\tau }}}$.

• Magnitude and direction of the torque: The magnitude of the torque depends on the magnitude of the lever arm, the magnitude of the force and the angle between these two vectors:
${\displaystyle \tau =\pm Fr\sin {\theta }}$


In special cases where the axis of rotation is given: The torque is positive when the torque causes the object to rotate in the counter-clockwise direction and negative when the torque causes the object to rotate in the clockwise direction.

• In general the torque is a vector product of the lever arm and the force:
${\displaystyle {\vec {\tau }}={\vec {r}}\times {\vec {F}}}$


### Static Equilibrium Revisited

• An object is in static equilibrium if and only if ${\displaystyle {\vec {\tau }}_{net}=0}$ and ${\displaystyle {\vec {F}}_{net}=0}$, where the net torque is the sum of all torques and the net force is the sum of all forces acting on an object.

• Static equilibrium problem solving strategy:

1. Determine all forces acting on an object and draw an arrow for each force indicating:
• A point where this force is applied
• In which direction this force acts
2. For each force:
• Identify the magnitude and the direction of lever arm
• Draw a diagram consisting of lever arm and force vectors and determine the angle between these two vectors
• Determine the direction of the torque and then compute its magnitude.
3. Substitute into the equation ${\displaystyle \Sigma \tau =0}$
4. Select the coordinate system
5. Draw a free-body diagram
6. Compute the components of the forces
7. Substitute into the equations ${\displaystyle \Sigma F_{x}=0}$ and ${\displaystyle \Sigma F_{y}=0}$
8. Solve the system of three equations using the method of substitution