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Wize University Physics Textbook (Master) > Angular / Rotational Motion

Torque

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Torque


Torque is simply tendency of rotation. It is a rotational equivalent of force and is defined as:


τ=∣r⃗ × F⃗∣=rFsin⁡θ\boxed{\tau=\left|\vec{r}\ \times\ \vec{F}\right|=r F\sin\theta}

  • where rris the magnitude of vector r⃗\vec{r}which is defined from center of rotation (also known as the reference point) to the point of exertion of force (also known as point of action).
  • θ\theta is the angle between r⃗\vec{r} and F⃗\vec{F}
  • unit for torque is N.mN.m







Watch Out!
Torque is a vector so it has magnitude and direction. In two dimensions, torque is positive if it generates a counter-clockwise rotation and it is negative if it generates a clockwise rotation.

Exam Tip
If there are more than one force acting on an object, net torque of the system would be the sum of all torque vector corresponding to different forces. Make sure to add them as Vectors!!

Example:





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Wize Tip
Forces cause NO torque when they are:
  1. Parallel to the axis of rotation,(sin⁡θ=0)(\sin\theta=0)
  2. Acting at the center of rotation (r=0)




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Example: Rotating a Door about its Hinge


You and your friend are trying to push open a gate that has a single hinge on one side. The gate starts opening if you apply a force of 60N perpendicular to the gate, at a point that is 60cm horizontally away from the hinge, parallel to the ground. If your friend is only able to push the gate with a force of 40N perpendicular to the door, where should they apply the force in order to open the gate?

Solution:

The same amount of torque is required to open the gate.

You

Since you apply a force of 60N perpendicular to the gate, the angle between r⃗\vec r and F⃗\vec F is 90°90\degree :
∣τ⃗∣=rFsin⁡(90°)|\vec \tau|=r F \sin(90\degree)
∣τ⃗∣=(0.60)(60)sin⁡(90°)|\vec \tau|=(0.60)(60)\sin(90\degree)
∣∣τ⃗∣∣=36 N.m||\vec \tau||=36 \ N.m

Your friend

∣τ⃗∣=rFsin⁡(90°)|\vec \tau|= r F\sin(90\degree)
36=r(40)sin⁡(90°)36= r(40)\sin(90\degree)
r=3640=0.9mr=\dfrac{36}{40}=0.9 m

Therefore, your friend should push the door at a distance of 90 cm away from the hinge, parallel to the ground.

Seven forces are applied to a beam supported at the center as shown. Rank the torques produced by each force about the hinge from greatest to least, labeling the forces from 1-7 starting left to right (2F forces make an angle of 45 °45\ \degreerespect to the beam)





Extra Practice
Torque on a Circular Current Loop
Practice: Torque on a Circular Current Loop
A circular loop of radius 10.5 cm and 40 turns has a current of 2.50 A flowing through the coil. A magnetic field of strength 3.4 T acts at an angle of 35º to the normal of the plane of the loop. What is the magnitude of the net torque acting on the loop?
Practice: Torque on a Circular Current Loop
A circular loop of radius 10.5 cm and 40 turns has a current of 2.50 A flowing through the coil. A magnetic field of strength 3.4 T acts at an angle of 35º to the normal of the plane of the loop. What is the magnitude of the net torque acting on the loop?
One end of a uniform 4.0 m-long rod of weight w{\small w} is supported by a cable at an triangle of θ = 37°{\small \theta\ =\ 37\degree}with the rod. The other end rests against a wall, where it is held by friction, as shown. The coefficient of static friction between the wall end the rod is μs = 0.50.{\small \mu_s\ =\ 0.50.} Determine the minimum distance x{\small x} from point A at which an additional weight w{\small w} (the same as the weight of the rod) can be hung without causing the rod to slip at point A. Enter your answer in meters to the appropriate number of sig figs.
Forces: Standing on a beam
A horizontal beam with a length of 8.0 meters and weight of 200 N is attached to a wall by a pin connection. Its far end is attached to the cable that makes angle of 53 degrees with the beam. A person of weight 600 N stands 2.0 meters away from the wall. What is the tension of the cable (T) and the magnitude of the force that wall applies on the beam (R), respectively?
Hint: The total reaction force is the sum of a normal force (perpendicular to the wall) and a friction force (parallel to the wall). So the net reaction force will be at an angle. (In this problem we are not ignoring friction at the wall). In this problem, there are a few ways to solve it; but you can just treat the reaction force as a single force that provides a single torque at an angle to find the answer.
Practice: Balanced beam
A uniform 90 kg beam is resting on two supports as shown below. The supports are 20 cm and 80 cm away from the two ends of beam. If the beam is 180 cm, what are the forces acting on the beam from supports A and B?
A 3.0 m ladder of mass 12 kg rests against a frictionless wall. The coefficient of friction between the ladder and the floor is 0.60. Assume there is no friction between the wall and the ladder. What is the smallest angle that the ladder can make with the ground if the ladder is to remain at rest? (Enter your answer in degrees)
Forces: Torque
A seesaw of length 5m is supported at the center. A child of mass 35kg is sitting on one end. A second child of mass 45kg comes to play. Where should the second child sit in order to balance the seesaw? Enter your answer in meters, measured from the pivot axis of the seesaw.
Find the net torque on the wheel below about the axle through Operpendicular to the page, taking a = 10.0 cm and b = 25.0 cm. (Note for clarification: the 12.0 N force is acting directly above the axis O, at an angle from that point.) Give your answer in units of Nm to the appropriate number of significant figures.
Calculate the net torque (magnitude and direction) on the beam through the point labelled O (i.e. an axis perpendicular to the page at point O) and (b) through an axis through C perpendicular to the page. Give your answers in Nm.
A mass-less string is wrapped several times around a Cylinder . If the free end of the string is held in place and the Cylinder is released form rest, the string unwinds and the Cylinder descends. What is the acceleration of the Cylinder ?
A thin uniform rod (length=1m, mass = 5kg) is pivoted about a horizontal, frictionless pin through one end of the rod. The rod is released when it makes an angle of 60°60\degree above the horizontal. What is the angular acceleration of the rod at the instant it is released? (pick the closest answer)
A 10 kg mass is hanging from a rope wrapped around a pulley. The falling mass causes the pulley to rotate. The mass of the pulley is 2 kg. What is the acceleration of the falling mass, and what is the tension of the rope?
Practice: Condition of Static Equilibrium
A 3.0 m ladder of mass 12 kg rests against a frictionless wall. The coefficient of friction between the ladder and the floor is 0.60. What is the smallest angle that the ladder can make with the ground if the ladder is to remain at rest? (in degrees) Neglect the friction between the ladder and the wall.
Is the force exerted by the tendon attached to the forearm to hold the forearm horizontal greater than, equal to, or less than the weight of the forearm?
If a man can push with a maximum force of 90 N, how far from the door's rotation axis must he apply this force if 110-Nm torque is needed to start rotating a revolving door?
A wooden beam of mass 150 kg and length 10m is suspended from a horizontal wall support at the left end and connected by a cable to the same wall at the right end. The cable makes an angle of 45º with the horizontal. A person of mass 42 kg stands on the beam 2.5 m from the left wall. A 25 kg child also rests on the beam, at 7m from the left wall. Calculate the tension in the rope and the force exerted by the wall on the beam.
Newton’s First Law does not apply to rotating objects
torque
Calculate the net torque (magnitude and direction) on the beam through the point labelled O (i.e. an axis perpendicular to the page at point O) and (b) through an axis through C perpendicular to the page. Give your answers in Nm.
Torque due to a Magnetic Field
A circular loop of radius 10.5 cm and 40 turns has a current of 2.50 A flowing through the coil. A magnetic field of strength 3.4 T acts at an angle of 35º to the normal of the plane of the loop. What is the magnitude of the net torque acting on the loop?