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## Class 11 Physics (India)

### Unit 11: Lesson 4

Torque and equilibrium

# Torque and equilibrium review

Review the concept of torque and how it is affected by the applied force and lever arm.

## Key terms

Term (symbol)Meaning
Torque (tau)Measure of the twisting action caused by a force that can cause an object to rotate about an axis. Vector quantity with SI units of start text, N, end text, dot, start text, m, end text.
Net torque (\Sigma, tau)Sum of all the torques on a system
Balanced systemWhen the net torque on a system is zero
Lever arm Perpendicular distance from the axis of rotation to where the force is applied. Vector quantity with SI units of start text, m, end text.
Pivot pointPoint that an object rotates around. Sometimes called the fulcrum or rotational axis.

## Equations

EquationSymbol breakdownMeaning in words
tau, equals, r, F, sine, theta, equals, r, start subscript, \perp, end subscript, Ftau is torque, F is applied force, r is the radius from the axis of rotation to the location where the force is exerted, and theta is the angle between F and r when these vectors are placed tail to tail.Torque is proportional to both the lever arm and the force component perpendicular to the lever arm.

## How to visualize the torque equation

A wrench produces a torque on a nut if a force is applied to it correctly (see Figure 1). The equation for torque is:
tau, equals, r, F, sine, theta
Figure 1. Variables of the torque equation shown for a wrench and nut. The nut’s center is the pivot point.
To produce a torque, the force F must be applied at some distance r away from the pivot point. Since only the perpendicular component F, start subscript, \perp, end subscript produces torque, the equation includes sine, theta (see Figure 2 below).
Figure 2. Components of the applied force F aligned to the lever arm.The perpendicular component is F, start subscript, \perp, end subscript and the parallel component is F, start subscript, \parallel, end subscript.

### The magnitude of the torque depends on:

• Applied force F: Larger forces increase torque.
• Angle between the force and lever arm theta: Directing a force perpendicular to the lever arm increases the torque.
An applied force can result in zero torque if there is no lever arm or the applied force is parallel to the lever arm (see Figure 3 and 4 below).
Figure 3. Lever arm: these applied forces result in no torque on the wrench because of no lever arm r.
Figure 4. Direction of force: these applied forces result in no torque on the wrench because the applied force is parallel to the lever arm.

## How to determine the direction of torque

The direction of rotation can be clockwise (cw) or counterclockwise (ccw). These terms refer to the movement of hands on a clock (see Figure 5). In physics, the counterclockwise direction is defined as positive and clockwise is negative for rotational variables.
Figure 5. The rotation of a clock’s hands is the reference for defining rotational direction. Counterclockwise is the positive rotation direction and clockwise is the negative direction.
For example, a torque that rotates an object counterclockwise is a positive torque (see figure 6 below).
Figure 6. An applied force that causes a positive counterclockwise torque.
A torque that rotates an object clockwise is a negative torque (see figure 7 below).
Figure 7. An applied force that causes a negative clockwise torque.

For deeper explanations of torque, see our video about torque and equilibrium.
To check your understanding and work toward mastering these concepts, check out our exercises:

## Want to join the conversation?

• why is a counter clockwise positive and clockwise negative torque?
• That is an arbitrary convention that we follow. Questions should normally give you the direction that is considered positive though, like how it was given here.
• Which mass is the mass used in the equation T = F x r?
Is force acceleration times the mass of the person pushing or is it the mass of the object being pushed?
• The mass of the object being pushed, because it is the object that is being focused on within the system. The weight of the person pushing the object is irrelevant
• so im stuck home because of c-19. and studying applied physics for my 3rd trimester is this what i start with to study it.
• I would think probably not. Im currently in AP Physics 1 and I would say it would be a good idea to start with 2d kinematics
• Can we balance torque on a point which is in accelerated motion?
• Also is there a video explaining what lever arm is or why we use it. I do not understand why there is the formula that torque=lever arm*force
why is this used? what is its advantage of finding torque this way?
• This is the definition of torque. If you know the magnitude of the force being applied and the distance from the axis of rotation it is applied, then by multiplying the two, you now know the torque. It is the distance from where the force is applied to the axis of rotation that we refer to as the lever arm. Hope this helps!
(1 vote)
• This was great and helpful, but quick question regarding equilibrium and torque. So the equilibrium here seemed to be of an object with a pivot point but what about examples like a hanging sign... I find this incredibly difficult to think about because let's say you have a hanging sign that sticks out from a wall via a rod both with respective masses and a cable from a building holding it. You are trying to calculate the tension in the cable.

You have to find the tension of this static equilibrium case -- I know that you have to balance the torques though they add up to 0, but I guess I am just confused about how to find the tension Force from the torques.
(1 vote)
• Think about it this way (I am going to assume that we know where the cable is attached to the sign):

Both the rod and sign have mass. So that means that they both have weight. This weight is a force that acts on their centers of mass. This means there is a torque on the sign because of gravity. There are two torques on the sign through gravity due to the rod's weight and the sign's weight, treat these like two different torques and add them together to get the total torque due to gravity. Now, since you know that the system is in equilibrium, this means that force of tension must exert a torque equal in magnitude to the total torque due to gravity. You can create an equation from this:

Tension * lever arm = Torque due to gravity.

Now just divide by the lever arm and you have your tension.

Note: In this example I assumed that the cable is perpendicular to the sign, if the cable is not perpendicular, you have to find the component of the tension force that is perpendicular to the sign using some trigonometry and replace the tension in the equation with this. Hope this helps!