Rotational dynamics (3.11.1)
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Inertia is the resistance an object has to changes in its linear motion, with mass serving as its quantitative measure. A greater mass indicates greater inertia, meaning a larger force is needed to alter the object’s linear motion.
The moment of inertia is a measure of how much an object resists changes in its rotational motion. The greater the moment of inertia of the object, the larger the force required to cause a change in the object’s rotation. Moment of inertia depends on the mass of the object and the distance of its mass distribution from the axis of rotation (i.e. the axis around which the object is rotating).
An example of this is a spinning disc, which rotates around an axis through its centre. The heavier the disc or the bigger its radius (assuming uniform mass distribution), the greater its moment of inertia.
For a point mass, the moment of inertia is calculated using:
Where:
- is the mass of the object, in kg,
- is the distance from the axis of rotation, in m, and
- is the moment of inertia, in
For an extended object, such as a disc or a hoop, the moment of inertia is calculated by summing all the individual moments of inertia of each point mass that constitute the entire object:
Two extended objects with the same mass and radius can have different moments of inertia because of how their mass is distributed around the axis of rotation.
For example, a solid sphere rotating about its centre has a moment of inertia of:
However, for a hollow sphere, the moment of inertia is:
For a system containing multiple objects, the moment of inertia about a given axis can be found for the entire system by summing the individual moments of inertia of all the objects about the same axis.
Question walkthrough
Finding total moment of inertia
Sums the moment of inertia of a disc, I=½MR², and a small object on its rim treated as a point mass, I=mr², to find the total moment of inertia about the central axis.
For a rotating object, the rotational kinetic energy is calculated by:
Where:
- is the moment of inertia, in and
- is the angular velocity, in
The equation for the rotational kinetic energy is the rotational analogue of the linear equation for kinetic energy:
Each quantity is converted from being a linear quantity to a rotational quantity – velocity is converted to angular velocity , and mass (inertia) is converted to moment of inertia
Question walkthrough
Finding rotational KE of a sphere
Uses I=(2/5)MR² for a solid sphere and E_K=½Iω² to find the rotational kinetic energy of a spinning sphere from its mass, radius, and angular velocity.
A flywheel is a wheel that has a high mass and so, a high moment of inertia. Due to its high moment of inertia, once a flywheel is spinning it is difficult to stop. The frictional force between the axis of rotation and the flywheel is minimised by using smooth materials and lubricant.
As the spinwheel spins faster (angular velocity, increases) it accumulates a large amount of rotational kinetic energy and is said to be charged. Enough power must be supplied to the flywheel to overcome the frictional forces and keep it spinning. When the flywheel reaches its maximum energy storage capacity as rotational kinetic energy, it is said to be fully charged.
Flywheels can provide additional energy to systems by decelerating and transferring some of their rotational kinetic energy to other components. Flywheel batteries are specifically designed to store as much energy as possible for later use.
The amount of energy a flywheel can store depends on its moment of inertia and its angular velocity.
- Moment of inertia: The greater the mass of the flywheel, the greater the moment of inertia and, so, the greater the rotational kinetic energy. Heavier flywheels can store more energy.
- Angular velocity: The greater the angular velocity of the flywheel, the faster it spins and, therefore, the greater the rotational kinetic energy. A faster-spinning flywheel can store more energy.
There is a limit on the mass and angular velocity of a flywheel:
- A flywheel with a very large mass will be large in size and impractical.
- At greater angular velocity centrifugal force is greater, and if too large can cause the flywheel to break.
To reduce energy losses due to friction flywheels are lubricated, or even levitated using superconducting magnets, to reduce friction between the wheel and its bearings. Placing flywheels in a vacuum can also eliminate air resistance.
In many systems, the power input and the power demand are not constant. If these fluctuations are not smoothed out, then the system may vibrate, operate unevenly, and the components will wear out faster.
Flywheels help stabilise such systems by ensuring that the angular momentum of rotating components remains constant. Any excess power charges the flywheel, which smoothly redistributes the stored energy to the rest of the system as needed.
- Load torque is the torque required to overcome the resistance of a load connected to a motor. This includes frictional, gravitational, and other resistive forces. When the load torque becomes too high, the flywheel slows down and releases energy to support the system.
- Engine torque is defined as the torque exerted by a machine’s engine. When the engine torque exceeds the load torque, the flywheel speeds up and stores the additional energy for future use.
There are advantages and disadvantages in the use of flywheels.
Flywheels are used in many everyday systems. Examples include:
- Regenerative brakes: Some cars contain flywheels. When the brakes are engaged, the flywheel is charged with the energy lost. Later, when the vehicle accelerates, the flywheel is engaged to help turn the wheels faster.
- Power grids: When a lot of electricity is used in a given area, the grid may not be able to keep up with demand. Flywheels can be used to provide additional power during peak times or to ease grid demand while backup power stations are starting up.
Angular displacement is defined as the angle through which an object or a point rotates about a fixed axis. It can be calculated using the equation:
Where:
- is the angular displacement (in ),
- is the arc length of the circular path (in ), and
- is the radius of the circular path (in ).
Angular velocity is a vector quantity defined as the rate of change of angular displacement with respect to time. It is calculated using:
Where:
- is the angular velocity (in
- is the linear speed (in
- is the radius of rotation (in
- is the angular displacement (in and
- is the time duration (in
One can define the angular acceleration as the rate of change of angular velocity:
Where:
- is the angular acceleration in ,
- is the tangential acceleration in ,
- is the radius of rotation in ,
- is the change in angular position in ,
- is the change in angular velocity in , and
- is the time duration in .
The translational equations of motion for constant linear acceleration can be rewritten in terms of rotational quantities to obtain the following rotational equations of motion:
| Translational | Rotational |
|---|---|
Where:
- is the final angular velocity,
- is the initial angular velocity,
- is the angular displacement,
- is the time duration, and
- is the angular acceleration.
Question walkthrough
Finding angular acceleration of spinning top
Uses α=(ω₂−ω₁)/t to find the angular acceleration of a spinning top from its initial and final angular velocities and the time taken.
When the angular velocity of a rotating object is constant (i.e. ), the equation relating angular displacement and time simplifies to
As a result, the graph of angular displacement against time illustrates a direct proportionality between and (a straight line):
Where:
- is the final angular velocity,
- is the initial angular velocity,
- is the angular displacement,
- is the time duration, and
- is the angular acceleration.
In rotational motion, the relationship between angular velocity, and time, depends on the behaviour of angular acceleration,
- When the angular velocity is constant, the graph of angular velocity against time is a straight horizontal line.
- If the angular acceleration is constant and non-zero, the graph of angular velocity against time for constant angular acceleration is a straight oblique line.
- However, if the angular acceleration is not constant, the graph of angular velocity against time is a parabola:
- concave up if is increasing
- concave down if is decreasing
When the angular acceleration is constant but unknown, its value can be found from the gradient of an angular velocity–time graph. Similarly, when it is not constant, angular acceleration can be found by drawing a tangent at a given instant and taking its gradient.
The area under the angular velocity–time graph gives the total angular displacement travelled.
Question walkthrough
Finding angular displacement from a graph
Uses the trapezium area under an angular velocity–time graph to find the total angular displacement of a flywheel that accelerates uniformly, then rotates at constant angular velocity.
Torque is a measure of the rotational force applied to an object. It quantifies the tendency of a force to cause an object to rotate about an axis or pivot point. Torque depends on the distance from the axis of rotation:
Where:
- is the torque
- is the applied force ( and
- is the perpendicular distance from the axis of rotation to the line of action of the force (
According to Newton’s second law of rotational motion, torque may be expressed in terms of the angular acceleration and moment of inertia:
Where:
- is the moment of inertia , and
- is the angular acceleration .
In linear systems, linear momentum is equal to the mass multiplied by the velocity. There is a rotational equivalent where the linear momentum becomes the angular momentum, the mass becomes the moment of inertia, and the velocity becomes the angular velocity.
Therefore, the angular momentum becomes:
Where:
- is the angular momentum
- is the moment of inertia and
- is the angular velocity
For a point mass, where the moment of inertia is the angular momentum is:
Since then the angular momentum for a point mass may be written as:
Where:
- is the mass , and
- is the linear velocity\)
The formula for angular momentum, shows that angular momentum is proportional to both the moment of inertia and the angular velocity.
- For the same angular velocity, objects with a greater mass and, therefore, greater moment of inertia have greater angular momentum (top illustration).
- For the same moment of inertia, objects with a greater angular velocity have greater angular momentum (bottom illustration).
When there are no external forces applied to the system, the total angular momentum between two events is constant (i.e. it is conserved):
Where and represent the initial and final states of the system. This is known as the law of conservation of angular momentum.
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For example, when an ice skater begins to spin, they initially extend their arms in front of them. As they pull their arms closer to their body, they start to spin faster with a greater angular velocity
Bringing their arms in reduces their moment of inertia, increasing their angular velocity to conserve angular momentum. The total angular momentum remains constant in both positions.
Question walkthrough
Finding angular velocity via momentum conservation
Uses conservation of angular momentum I_iω_i=I_fω_f to find an ice skater’s angular velocity after tucking in her arms reduces her moment of inertia.
The angular impulse is defined as the change in angular momentum. The most common units of angular impulse are and :
The torque is defined as the rate of change of angular momentum. Hence,
If the torque on the system is constant, then the previous equality may be written as:Moreover, as :
Where:
- is the moment of inertia
- is the angular velocity
- is the torque and
- is the time the torque is applied
Just as linear impulse is the area under a force–time graph (integrating force over time), angular impulse is found by integrating torque over time. Therefore, when torque varies, the angular impulse is the area under the torque – time graph. This area also represents the change in angular momentum.
Question walkthrough
Finding angular impulse from a graph
Uses the trapezium area under a torque–time graph to find the angular impulse delivered to an object between two given time points.
Question walkthrough
Finding angular velocity from angular impulse
Uses the angular impulse–momentum theorem τΔt=IΔω to find a lever’s angular velocity after a constant torque acts on it, converting the result from rad/s to rev/s.
Work is required to rotate an object. In linear systems, the work done is equal to the force multiplied by the distance. In rotational systems, the equivalent of force is the torque, and the equivalent of distance is the angular displacement.
Where:
- is the work done
- is the torque and
- is the angular displacement
Power is defined as the rate at which work is done, or energy is transferred over time. It quantifies how quickly work is performed.
The power for rotational systems can be expressed by substituting in the equation for work, Hence, the power becomes:
Since then
Where:
- is the power
- is the torque and
- is the angular velocity
In practical applications, it is essential to consider friction. Rotating systems encounter frictional torque, which requires a portion of the system’s power to overcome.
Question walkthrough
Finding power lost to friction
Uses τ_net=Iα to find the net torque on a flywheel, subtracts this from the applied torque to find the frictional torque, then P=τω to find the power dissipated by friction.