Rot Dy 1
Dynamics of Rotational Motion
Torque
Torque is the rotational counterpart for force. It is the measure of the tendency of a force to cause or
change the rotational motion of a body. Recall that force can be described as a “push” or a “pull” and
that it affects the linear acceleration of an object. Similarly, we can describe torque as a “twist” or a
“turn”, and it affects the angular acceleration of an object.
⃗ acts at the edge of the wheel at an angle 𝜃 from the horizontal,
Consider a wheel of radius r. A force 𝑭
making the wheel rotate.
⃗ be the vector from the axis of rotation (in this case, the
Let 𝒓
centre of the wheel) to the point where the force acts. From
the figure, we can see that the component of the force that is
⃗ does not contribute to the rotation of the wheel
parallel to 𝒓
– only the perpendicular component makes the wheel rotate.
The magnitude of the torque is given by
|𝜏| ≡ 𝑟 ∙ 𝐹⊥ = 𝑟𝐹 𝑠𝑖𝑛 𝜃. (10.1)
where r is the “lever arm” (distance from axis of rotation to
the point of application of force), and 𝐹⊥ is the component of
the force that is perpendicular to the lever arm. The unit of torque is [𝜏] =[ 𝑟][ 𝐹] = m∙N.
For fixed axis, torque has a sign (+ or –) :
Positive torque causes counterclockwise CCW rotation.
Negative torque causes clockwise (CW) rotation.
Like force, torque is a vector – it has a direction. The vector for of torque is given by the cross product
⃗ = 𝒓
𝝉 ⃗.
⃗ ×𝑭 (10.2)
The magnitude of the torque is given by equation (10.1). What about the direction? The direction of the
torque is perpendicular to both the lever arm and the force and is determined by the right-hand rule.
Torque and Angular Acceleration for a Rigid Body
The torque can also be written in terms of the moment of inertia of a rigid body:
∑𝝉 ⃗⃗
⃗ = 𝐼∙𝜶 (10.3)
where I is the moment of inertia and 𝜶 ⃗⃗ is the angular acceleration. Note that the above equation is
similar to Newton’s second law (recall that 𝝉 ⃗ is the rotational equivalent of ⃗𝑭, 𝐼 is the rotational
⃗⃗ is the rotational equivalent of 𝒂
equivalent of 𝑚, and 𝜶 ⃗ ). Thus, equation (10.3) is the rotational analogue
of Newton’s second law for a rigid body.
Notes:
1. In using the above equation, the pivot used in calculating the torque should also be the
same pivot used to calculate the moment of inertia.
2. In using the above equation, 𝒂 ⃗ should be in rad/s2 !!!
3. Only external torques can affect the angular acceleration of a rigid body.
4. Sum of internal torques (caused by internal forces) is ZERO.
5. Conditions for validity of the above equation:
SEF005 Queen Mary University of London
Mechanics and Materials School of Physics and Astronomy
Rot Dy 2
rotating body is rigid
moment of inertia is constant
axis of rotation is not changing direction
Rigid-Body Rotation About a Moving Axis
When an object has both translational and rotational motion, its kinetic energy is the sum of the
translational and rotational kinetic energies of the object:
𝐾 = 𝐾𝑡𝑟𝑎𝑛𝑠 + 𝐾𝑟𝑜𝑡 = 12𝑀𝑣𝐶𝑀 2 + 12𝐼𝐶𝑀 𝜔2 (10.4)
where 𝑀 = mass of the body
𝑣𝐶𝑀 = speed of the centre of mass of the body
𝐼𝐶𝑀 = moment of inertia about axis through the centre of mass
𝜔 = angular speed of the body
Angular Momentum
Returning to the Newton’s second law equation (10.3), we can find the equation of angular momentum
𝑑𝝎
⃗⃗⃗
⃗⃗ =
by substituting 𝜶 :
𝑑𝑡
𝑑𝝎
⃗⃗⃗ 𝑑
∑𝝉 ⃗⃗ = 𝐼 ∙
⃗ = 𝐼∙𝜶
𝑑𝑡
= 𝑑𝑡
(𝐼 𝝎
⃗⃗⃗ ). (10.5)
Comparing the above equation to the relationship between net force and linear momentum (i.e. net
force is the time rate of change of the linear momentum), we can define angular momentum as
⃗ = 𝐼𝝎
𝑳 ⃗⃗⃗ . (10.6)
Thus, the time rate of change of the angular momentum of a rigid body is the torque of the net force acting on it.
For a particle, the angular momentum is given by
⃗𝑳 = 𝒓
⃗ ×𝒑
⃗ = 𝑚𝒓
⃗ ×𝒗
⃗ (10.7)
⃗ = 𝑚𝒗
where 𝒑 ⃗ = linear momentum of the particle
⃗ = position vector of particle
𝒓
𝑚 = mass of the particle
⃗ = velocity of the particle
𝒗
SEF005 Queen Mary University of London
Mechanics and Materials School of Physics and Astronomy
