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Torque and Rotational Motion Tutorial Physics

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Torque and Rotational Motion Tutorial
What is Torque?
Torque is a measure of how much a force acting on an object causes that object to rotate. The object rotates
about an axis, which we will call the pivot point, and will label 'O'. We will call the force 'F'. The distance from
the pivot point to the point where the force acts is called the moment arm, and is denoted by 'r'. Note that
this distance, 'r', is also a vector, and points from the axis of rotation to the point where the force acts. (Refer
to Figure 1 for a pictoral representation of these definitions.)
Figure 1: Definitions
Torque is defined as Γ = r × F = rFsin(θ).
In other words, torque is the cross product between the distance vector (the distance from the pivot point
to the point where force is applied) and the force vector, 'a' being the angle between r and F.
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Cross Product
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Using the right hand rule, we can find the direction of the torque vector. If we put our fingers in the direction
of r, and curl them to the direction of F, then the thumb points in the direction of the torque vector.
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Right Hand Rule
Imagine pushing a door to open it. The force of your push (F) causes the door to rotate about its hinges (the
pivot point, O). How hard you need to push depends on the distance you are from the hinges (r) (and several
other things, but let's ignore them now). The closer you are to the hinges (i.e. the smaller r is), the harder it is
to push. This is what happens when you try to push open a door on the wrong side. The torque you created
on the door is smaller than it would have been had you pushed the correct side (away from its hinges).
Note that the force applied, F, and the moment arm, r, are independent of the object. Furthermore, a force
applied at the pivot point will cause no torque since the moment arm would be zero (r = 0).
Another way of expressing the above equation is that torque is the product of the magnitude of the force
and the perpendicular distance from the force to the axis of rotation (i.e. the pivot point).
Let the force acting on an object be broken up into its tangential (F tan) and radial (F rad) components (see
Figure 2). (Note that the tangential component is perpendicular to the moment arm, while the radial
component is parallel to the moment arm.) The radial component of the force has no contribution to the
torque because it passes through the pivot point. So, it is only the tangential component of the force which
affects torque (since it is perpendicular to the line between the point of action of the force and the pivot
point).
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Pivot Point
Figure 2: Tangential and radial components of force F
There may be more than one force acting on an object, and each of these forces may act on different point on
the object. Then, each force will cause a torque. The net torque is the sum of the individual torques.
Rotational Equilibrium is analogous to translational equilibrium, where the sum of the forces are equal to
zero. In rotational equilibrium, the sum of the torques is equal to zero. In other words, there is no net
torque on the object.
∑τ = 0
Note that the SI units of torque is a Newton-metre, which is also a way of expressing a Joule (the unit for
energy). However, torque is not energy. So, to avoid confusion, we will use the units N.m, and not J. The
distinction arises because energy is a scalar quanitity, whereas torque is a vector.
Here is a useful and interesting interactive activity on rotational equilibrium.
Torque and Angular Acceleration
In this section, we will develop the relationship between torque and angular acceleration. You will need to
have a basic understanding of moments of inertia for this section.
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Moment of Inertia
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Figure 3: Radial and Tangential Components of Force, two dimensions
Imagine a force F acting on some object at a distance r from its axis of rotation. We can break up the force
into tangential (F tan), radial (F rad) (see Figure 3). (This is assuming a two-dimensional scenario. For three
dimensions -- a more realistic, but also more complicated situation -- we have three components of force:
the tangential component F tan, the radial component F rad and the z-component F z. All components of force
are mutually perpendicular, or normal.)
From Newton's Second Law, F tan = ma tan
However, we know that angular acceleration, α , and the tangential acceleration atan are related by:
a tan = rα
Then,
F tan = mrα
If we multiply both sides by r (the moment arm), the equation becomes
F tanr = mr 2α
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Note that the radial component of the force goes through the axis of rotation, and so has no contribution to
torque. The left hand side of the equation is torque. For a whole object, there may be many torques. So the
sum of the torques is equal to the moment of inertia (of a particle mass, which is the assumption in this
derivation), I = mr 2 multiplied by the angular acceleration, α.
∑τ = I ⋅ α
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Panel 4: Radial, Tangential and z-Components of Force, three dimensions
If we make an analogy between translational and rotational motion, then this relation between torque and
angular acceleration is analogous to the Newton's Second Law. Namely, taking torque to be analogous to
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force, moment of inertia analogous to mass, and angular acceleration analogous to acceleration, then we
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have an equation very much like the Second Law.
Example Problem: The Swinging Door
Question
In a hurry to catch a cab, you rush through a frictionless swinging door and onto the sidewalk. The force you
extered on the door was 50N, applied perpendicular to the plane of the door. The door is 1.0 m wide.
Assuming that you pushed the door at its edge, what was the torque on the swinging door (taking the hinge
as the pivot point)?
Hints
1. Where is the pivot point?
2. What was the force applied?
3. How far from the pivot point was the force applied?
4. What was the angle between the door and the direction of force?
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Solution
Physics Tutorials
Home (/outreach/physics-tutorials)
Torque Self-Test
The Right Hand Rule (/torque-self-test-right-hand-rule)
Definition of Torque (/torque-self-test-definition-torque)
Net Torque (/torque-self-test-net-torque)
Angular Acceleration (/torque-self-test-angular-acceleration)
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