Note: This is an ADVANCED HIGHER LEVEL topic. Although all may learn from it, only HL students will be tested on its content.
IB recommended length: 7 hours
At the end of topic A.2 we looked at circular motion, including concepts such as centripetal acceleration and centripetal force. In this AHL topic we will look into more detail about the topic of torque and rotating rigid bodies, where the whole body is rotating, rather than a body moving in a circle about a point. We will look at the factors that influence this motion and let us make predictions about rotational motion. in particular the distribution of mass in the rotating system.
If an object spins or rotates about an axis or pivot, then we can describe its motion using angular terminology.
Angular displacement, θ: the change in the angular position of the object, θ (the lower-case Greek letter theta), given the symbol ∆θ, and measured in units of radians (preferably) or degrees. Angular displacement = final angle − initial angle, ∆θ = θf − θi. Angular displacement is the equivalent of position in linear quantities.
Angular velocity, ω: the same concept as introduced as part of circular motion in topic A.2. Angular velocity, also known as angular speed ω (the lower-case Greek letter omega), is the rate of change of the angle θ, in units of radians per second or degrees per second. Angular velocity is the equivalent of linear velocity in linear quantities.
Angular acceleration, α: the rate of change of angular velocity. The symbol of angular acceleration is the lower-case Greek letter alpha, α, given in units of radians per second squared, rad / s². Angular acceleration is the equivalent of linear acceleration in linear quantities.
These equations are the angular equivalent of the suvat equations for linear motion. They are used, in situations of uniform angular acceleration, α, measured in rad / s², to predict a body's angular position θ (measured in radians), angular displacement ∆θ (also measured in radians), and angular speed ω (measured in radians per second, rad/s) and time t (measured in seconds). The subscripts f and i are used to indicate final and initial values.
∆θ = (ωf + ωi)/2 × t
In English, the change in angular position is equal to the average angular velocity multiplied by the time taken.
ωf = ωi + αt
In English, the final angular velocity is the initial angular velocity plus the angular acceleration multiplied by the time taken.
∆θ = ωi.t + 1/2 × αt²
ωf² = ωi² + 2α∆θ
Torque, symbol τ, the lower-case Greek letter tau, is a force that causes rotational motion. An unbalanced torque applied to an extended rigid body will cause angular acceleration, changing the angular velocity, ω (in radians per second) of the rotation.
The magnitude of a torque in measured in newton metres (note this is the same as joules, but we never use the word joules to describe the unit of torque). This unit comes from the equation that allows to calculate torque:
τ = F.r sin θ
where F is the applied force, measured in newtons, r is the straight-line distance from the pivot or axis of rotation (the point about which the object is rotating) to the point of application of the force, measured in metres. θ is the angle between the direction of application of the force, and the line of r, the straight line connecting the pivot to the point of application of the force. When θ shows a right angle (π/2 radians or 90º) then sin θ = 1 and the torque is at a maximum.
A consequence of this equation in use is that identical torques can be produced by a large force acting close to the axis of rotation, or a smaller force applied far from the axis of rotation. To create a maximum torque you need to apply as large a force as possible, as far as possible from the axis of rotation.
Multiple torques can act on a body at the same time. If positive torques indicate one direction (clockwise or anticlockwise) then negative torques indicate the opposite direction. Given the information about the direction of the torques then finding the total torque on an object is simply adding them all together.
If the total torque on an object zero then it is in rotational equilibrium. An object in rotational equilibrium has no angular acceleration. If the object is not rotating it won't start rotating. If it is already rotating it will continue rotating with the same angular velocity.
The moment of inertia, I, of a rotating rigid body is roughly equivalent to the kinetic energy of a body in linear motion. The magnitude of the moment of inertia depends on the distribution of mass of the rotating object. If two discs, both rotating about an axis in their centre, have the same diameter and the same mass, but disc A has an even distribution of mass, while disc B has more mass concentrated on the edge of the disc, then disc B will have a greater moment of inertia.
If, instead of a disc or another continuous rigid body, you have a system of point masses (imagine a collection of small spherical masses conected by light (i.e. massless) rods, rotating about a common axis) the the moment of inertia of the system is calculated using the equation:
I = Σmr²
where I is the moment of inertia, with units of kg m². The upper-case Greek letter sigma, Σ, represents the mathematical operation of "sum of". For each of the point masses, m is the mass of the point mass in kg and r is the straight-line distance from the axis to the point mass in metres. So to calculate the moment of inertia we must find the product of the mass and the axis-distance squared for each point mass, and then sum those products for every point mass in the system.
The subject guide gives no other equations for moment of inertia so I don't think they will be examined on, but the following equations give the moment of inertia for several simple shapes.
Ring: for a thin ring of radius R (in m) and mass M (in kg) with an axis of rotation through the middle of the ring and perpendicular to the plane of the ring. I = MR².
Disc or cylinder: for a uniform disc or cylinder of radius R (in m) and mass M (in kg) with an axis of rotation through the centre of the disc or cylinder and perpendicular to the plane of the cross-section of the disc or cylinder. I = 1/2 × MR².
Sphere: for a uniform sphere of radius R (in m) and mass M (in kg) with an axis of rotation through the centre of the sphere. I = 2/5 × MR².
Rod: a uniform thin rod of length L (in m) and mass M (in kg) with an axis of rotation through the centre of the rod and perpendicular to the length of the rod. I = 1/12 × MR².
Newton's second law for linear motion relates the rate of change of momentum of an object to the force exerted on it, or, for constant mass, the force and the acceleration. For a rotating rigid body, the moment of inertia takes the place of the mass and the torque takes the place of the force.
τ = Iα
or
α = τ / I
where τ is the torque in N m, I is the moment of inertia in kg m² and α is the angular acceleration in rad/s².
If the angular equivalent of kinetic energy is the moment of inertia, then the angular equivalent of momentum is angular momentum, symbol L and units kg m/s².
The angular momentum, L in kg m/s², for a rotating body is the product of the moment of inertia, I in kg m² and the angular velocity, ω in rad/s.
L = Iω
Any body rotating with constant angular velocity will have a constant angular momentum. But, if a resultant torque acts on the body then it will experience angular acceleration and the angular momentum will change.
The change in angular momentum caused by the application of a resultant torque is called the angular impulse, ∆L.
∆L = τ∆t = ∆(Iω)
where ∆L is the angular impulse in kg m/s², τ is the torque in N m, ∆t is the time taken in seconds, I is the moment of inertia in kg m² and ω is the angular velocity in rad/s.
We have previously said that the moment of inertia for a rotating rigid body is analagous to the the kinetic energy of an object in linear motion, but rotating rigid bodies do actually have a kinetic energy as well. For example, a chemical or an elastic potential energy could be used to rotate a rigid body with no loses.
The equation for the kinetic energy of a rotating rigid body is:
Ek = 1/2 × Iω² = L² / 2I
where Ek is the kinetic energy in joules, I is the moment of inertia in kg m², ω is the angular velocity in rad/s and L is the angular momentum in in kg m/s².
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