Define angular displacement .

Angular displacement is the change in angle through which a rigid body has rotated relative to a fixed point, measured in radians .

Angular acceleration is the rate of change of .......... with respect to time.

Angular acceleration is the rate of change of angular velocity with respect to time.

On an angular velocity–time graph , what do the gradient and the area under the graph represent?

Gradient = angular acceleration Area under the graph = angular displacement

True or False? Angular acceleration and centripetal acceleration are the same thing.

False. Angular acceleration describes the rate of change of angular velocity, whereas centripetal acceleration is the acceleration directed towards the centre of a circular path; the two are distinct quantities.

In the rotational kinematic equations, angular displacement θ is the rotational equivalent of linear .......... .

In the rotational kinematic equations, angular displacement θ is the rotational equivalent of linear displacement , s .

True or False? The rotational kinematic equations can be applied when angular acceleration is not constant.

False. Like their linear counterparts, the rotational kinematic equations only apply for uniform (constant) angular acceleration .

Why does a couple produce a zero resultant force but a non-zero resultant torque ?

The two forces in a couple are equal in magnitude and opposite in direction, so they cancel to give a resultant force of zero. However, both forces act in the same rotational sense about the pivot, so their torques add .

True or False? A moment and a turning force are the same thing.

False. A moment is the effect that a turning force has on a system when applied at a distance from a pivot, not the force itself.

When calculating the resultant torque about a chosen point, how can you simplify the calculation if some forces have unknown magnitude?

Choose a point through which the line of action of the unknown forces passes — this makes their torque about that point zero .

The torque of a force is equal to the applied force multiplied by the .......... distance between the axis of rotation and the line of action of the force.

The torque of a force is equal to the applied force multiplied by the perpendicular distance between the axis of rotation and the line of action of the force.

Define moment of inertia .

The moment of inertia of a rigid, extended body is its resistance to a change of rotational motion, depending on the distribution of mass around a chosen axis of rotation. It is measured in kg m 2 .

Give two factors that affect the moment of inertia of a body.

The total mass of the body How the mass is distributed relative to the axis of rotation

What is the linear equivalent of moment of inertia?

Mass . Mass measures inertia (resistance to a change in linear motion); moment of inertia measures the equivalent resistance to a change in rotational motion.

True or False? A single rigid object has only one possible value for its moment of inertia.

False. The moment of inertia of an object depends on the chosen axis of rotation , so the same object can have a different moment of inertia for different orientations of that axis.

As a diver tucks their legs in, their mass is distributed over a .......... distance from the axis of rotation, which .......... their moment of inertia.

As a diver tucks their legs in, their mass is distributed over a smaller distance from the axis of rotation, which decreases their moment of inertia.

Torque is the rotational equivalent of .......... , and angular acceleration is the rotational equivalent of .......... .

Torque is the rotational equivalent of force , and angular acceleration is the rotational equivalent of (linear) acceleration .

True or False? The moment of inertia of a rotating body describes its resistance to changes in linear velocity.

False. Moment of inertia describes resistance to changes in angular (rotational) velocity, not linear velocity — this is what mass describes in linear motion.

Define the principle of conservation of angular momentum .

The angular momentum of a system always remains constant, unless a net torque acts on the system.

Give three real-world examples where conservation of angular momentum causes a change in rotational speed.

A person on a spinning chair speeds up as their arms and legs are contracted Ice skaters spin faster by contracting their arms and legs Objects in elliptical orbits travel faster nearer the object they orbit Tornadoes spin faster as their radius decreases

When an ice skater pulls their arms in, their moment of inertia .......... and, since angular momentum is conserved, their angular velocity must .......... .

When an ice skater pulls their arms in, their moment of inertia decreases and, since angular momentum is conserved, their angular velocity must increase .

True or False? An object moving in a straight line cannot have angular momentum.

False. An object travelling in a straight line can still have angular momentum about a chosen axis, depending on its position relative to that axis.

What are the SI units of angular impulse?

kg m 2 s -1 (equivalent to N m s).

How can angular impulse be determined from a torque–time graph?

It is equal to the area under the graph .

A resultant torque can be defined as the rate of change of .......... .

A resultant torque can be defined as the rate of change of angular momentum .

On a torque–time graph, the resultant torque acts in the negative direction for part of the interval. What effect does this have on the total angular impulse?

It contributes a negative angular impulse , which reduces the overall change in angular momentum compared with the positive contributions.

Define frictional torque .

The difference between the applied torque and the resulting net (observed) torque .

How can the work done by a rotating torque be found from a graph?

It is equal to the area under a torque–angular displacement graph .

Net torque = applied torque .......... frictional torque.

Net torque = applied torque minus frictional torque.

True or False? The angular displacement θ in W = τθ and P = τω can be measured in degrees or radians.

False. θ (and angular velocity ω ) must always be in radians for these equations to give the correct work done or power.

Why is frictional torque generally minimised in rotating machinery?

Because power must be expended to overcome it, wasting energy as it is transferred to heat and sound .

When an object slips (slides) rather than rolls, its angular velocity is .......... .

When an object slips (slides) rather than rolls, its angular velocity is zero .

Define the function of a flywheel.

To act as an energy reservoir , storing and supplying rotational kinetic energy when required.

How does a flywheel smooth out fluctuations in engine torque?

It speeds up or slows down due to its inertia, absorbing sharp fluctuations so the torque delivered is smoother.

Give three ways friction can be reduced in a flywheel system.

Lubricating the bearings Using superconducting bearings so the flywheel levitates Operating the flywheel in a vacuum or sealed container

In conventional braking, the kinetic energy store of a vehicle is transferred as waste to the .......... energy store.

In conventional braking, the kinetic energy store of a vehicle is transferred as waste to the thermal energy store.

In a production process such as riveting, how does combining an electric motor with a flywheel benefit the machine?

The motor charges up the flywheel, which then delivers a short burst of energy for the process. This prevents the motor from stalling , so a less powerful motor can be used.

A LevelPhysicsAQAFlashcardsEngineering PhysicsRotational Dynamics

Rotational Dynamics (AQA A Level Physics): Flashcards

Exam code: 7408

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FrontRotational Motion
Define angular displacement.

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Cards in this collection (71)

Define angular displacement.
Angular displacement is the change in angle through which a rigid body has rotated relative to a fixed point, measured in radians.
Define angular velocity.
Angular velocity is the rate of change of angular displacement with respect to time, measured in rad s^-1: $ω = \frac{Δ θ}{Δ t}$
Define angular acceleration.
Angular acceleration is the rate of change of angular velocity with respect to time, measured in rad s^-2: $α = \frac{Δ ω}{Δ t}$
How is linear speed v related to angular speed ω?
$v = r ω$ where r is the distance from the axis of rotation.
Angular acceleration is the rate of change of .......... with respect to time.
Angular acceleration is the rate of change of angular velocity with respect to time.
On an angular velocity–time graph, what do the gradient and the area under the graph represent?
- Gradient = angular acceleration
- Area under the graph = angular displacement
True or False?
Angular acceleration and centripetal acceleration are the same thing.
False.
Angular acceleration describes the rate of change of angular velocity, whereas centripetal acceleration is the acceleration directed towards the centre of a circular path; the two are distinct quantities.
How is angular acceleration α related to linear acceleration a?
$a = r α$ where r is the distance from the axis of rotation.
State the equation for final angular velocity, ω₂, in terms of initial angular velocity, angular acceleration and time.
$ω_{2} = ω_{1} + α t$
State the equation for angular displacement Δθ when ω₁, ω₂ and α are known, but not t.
${(ω_{2})}^{2} = {(ω_{1})}^{2} + 2 α Δ θ$
In the rotational kinematic equations, angular displacement θ is the rotational equivalent of linear ...........
In the rotational kinematic equations, angular displacement θ is the rotational equivalent of linear displacement, s.
What is the rotational equivalent of linear acceleration, a?
Angular acceleration, α.
Before using the rotational kinematic equations with a value given in RPM (revolutions per minute), what conversion must be carried out?
Convert the value to rad s^-1 using $ω = 2 π f$ , where $f = \frac{R P M}{60}$
True or False?
The rotational kinematic equations can be applied when angular acceleration is not constant.
False.
Like their linear counterparts, the rotational kinematic equations only apply for uniform (constant) angular acceleration.
Define torque.
Torque is the turning effect of a force about an axis, equal to the product of the applied force and the perpendicular distance from the axis of rotation to the line of action of the force: $τ = F r$
State the equation for the torque of a couple, in terms of a single force F and the perpendicular distance r.
$τ = 2 F r$
Why does a couple produce a zero resultant force but a non-zero resultant torque?
The two forces in a couple are equal in magnitude and opposite in direction, so they cancel to give a resultant force of zero. However, both forces act in the same rotational sense about the pivot, so their torques add.
True or False?
A moment and a turning force are the same thing.
False.
A moment is the effect that a turning force has on a system when applied at a distance from a pivot, not the force itself.
When calculating the resultant torque about a chosen point, how can you simplify the calculation if some forces have unknown magnitude?
Choose a point through which the line of action of the unknown forces passes — this makes their torque about that point zero.
The torque of a force is equal to the applied force multiplied by the .......... distance between the axis of rotation and the line of action of the force.
The torque of a force is equal to the applied force multiplied by the perpendicular distance between the axis of rotation and the line of action of the force.
A steering wheel is turned by a couple of equal and opposite forces. Why does the wheel not undergo linear acceleration?
The resultant force from the couple is zero, so by Newton's second law ( $F = m a$ ), there is no linear acceleration — the wheel only rotates.
Define moment of inertia.
The moment of inertia of a rigid, extended body is its resistance to a change of rotational motion, depending on the distribution of mass around a chosen axis of rotation. It is measured in kg m².
State the equation for the moment of inertia of a point mass.
$I = m r^{2}$
How is the total moment of inertia of an extended object calculated from its individual particles?
By summing mr² for all the particles that make up the body: $I = \sum m r^{2}$
Give two factors that affect the moment of inertia of a body.
- The total mass of the body
- How the mass is distributed relative to the axis of rotation
What is the linear equivalent of moment of inertia?
Mass. Mass measures inertia (resistance to a change in linear motion); moment of inertia measures the equivalent resistance to a change in rotational motion.
True or False?
A single rigid object has only one possible value for its moment of inertia.
False.
The moment of inertia of an object depends on the chosen axis of rotation, so the same object can have a different moment of inertia for different orientations of that axis.
As a diver tucks their legs in, their mass is distributed over a .......... distance from the axis of rotation, which .......... their moment of inertia.
As a diver tucks their legs in, their mass is distributed over a smaller distance from the axis of rotation, which decreases their moment of inertia.
Define Newton's second law for rotation.
For a rotating body, the torque required to give it a certain angular acceleration depends on its moment of inertia: $τ = I α$
Give the linear equivalent of the rotational equation $τ = I α$ .
$F = m a$
What is the rotational equivalent of mass?
Moment of inertia, I.
Torque is the rotational equivalent of .........., and angular acceleration is the rotational equivalent of ...........
Torque is the rotational equivalent of force, and angular acceleration is the rotational equivalent of (linear) acceleration.
A block hangs from a string wound around a pulley. As the block falls, which two Newton's second law equations must be applied to find the block's acceleration?
- Linear Newton's second law applied to the block: $m g - T = m a$
- Rotational Newton's second law applied to the pulley: $T R = I α$
True or False?
The moment of inertia of a rotating body describes its resistance to changes in linear velocity.
False.
Moment of inertia describes resistance to changes in angular (rotational) velocity, not linear velocity — this is what mass describes in linear motion.
Starting from $τ = F r$ and $a = r α$ , show how $τ = I α$ is derived for a point mass.
Substituting $F = m a$ and $a = r α$ into $τ = F r$ gives $τ = r (m r α) = (m r^{2}) α = I α$ , since $I = m r^{2}$ for a point mass.
Define angular momentum.
Angular momentum, L, is the rotational equivalent of linear momentum ( $p = m v$ ): $L = I ω$ , measured in kg m² rad s^-1.
State the equation for the angular momentum of a point mass, in terms of mass m, distance r and linear velocity v.
$L = m v r$
Define the principle of conservation of angular momentum.
The angular momentum of a system always remains constant, unless a net torque acts on the system.
Give three real-world examples where conservation of angular momentum causes a change in rotational speed.
- A person on a spinning chair speeds up as their arms and legs are contracted
- Ice skaters spin faster by contracting their arms and legs
- Objects in elliptical orbits travel faster nearer the object they orbit
- Tornadoes spin faster as their radius decreases
When an ice skater pulls their arms in, their moment of inertia .......... and, since angular momentum is conserved, their angular velocity must ...........
When an ice skater pulls their arms in, their moment of inertia decreases and, since angular momentum is conserved, their angular velocity must increase.
True or False?
An object moving in a straight line cannot have angular momentum.
False.
An object travelling in a straight line can still have angular momentum about a chosen axis, depending on its position relative to that axis.
Write the equation used to solve problems involving a change in a system's angular momentum, defining each term.
$I_{i} ω_{i} = I_{f} ω_{f}$
where I_i and ω_i are the initial moment of inertia and angular velocity, and I_f and ω_f are the final values.
Define angular impulse.
Angular impulse is the change in angular momentum produced by an average resultant torque τ acting for a time Δt, given by $Δ L = τ Δ t = Δ (I ω)$ .
What are the SI units of angular impulse?
kg m² s^-1 (equivalent to N m s).
How can angular impulse be determined from a torque–time graph?
It is equal to the area under the graph.
A resultant torque can be defined as the rate of change of ...........
A resultant torque can be defined as the rate of change of angular momentum.
True or False?
A small torque acting over a long time can produce the same angular impulse as a large torque acting over a short time.
True.
Angular impulse is the product of torque and time ( $Δ L = τ Δ t$ ), so a small torque over a long duration can equal a large torque over a short duration.
What condition must be satisfied to use $Δ L = τ Δ t$ directly, without averaging?
The resultant torque must be constant. If the torque varies, an average value must be used.
On a torque–time graph, the resultant torque acts in the negative direction for part of the interval. What effect does this have on the total angular impulse?
It contributes a negative angular impulse, which reduces the overall change in angular momentum compared with the positive contributions.
Define the work done by a rotating object.
$W = τ θ$ , the product of the torque and the angular displacement (in radians).
Define frictional torque.
The difference between the applied torque and the resulting net (observed) torque.
How can the work done by a rotating torque be found from a graph?
It is equal to the area under a torque–angular displacement graph.
State the equation linking power, torque and angular velocity.
$P = τ ω$
Net torque = applied torque .......... frictional torque.
Net torque = applied torque minus frictional torque.
True or False?
The angular displacement θ in W = τθ and P = τω can be measured in degrees or radians.
False.
θ (and angular velocity ω) must always be in radians for these equations to give the correct work done or power.
Why is frictional torque generally minimised in rotating machinery?
Because power must be expended to overcome it, wasting energy as it is transferred to heat and sound.
Define rotational kinetic energy.
The kinetic energy a rotating body possesses due to its angular velocity, given by $E_{k} = \frac{1}{2} I ω^{2}$ .
State two equations for rotational kinetic energy in terms of moment of inertia.
$E_{k} = \frac{1}{2} I ω^{2}$ and $E_{k} = \frac{L^{2}}{2 I}$ (where L is angular momentum).
What is the linear velocity of the point on a rolling object (rolling without slipping) that is in contact with the surface?
Zero.
What is the velocity of the topmost point of an object rolling without slipping, in terms of its centre-of-mass velocity v?
2*v (equivalently 2ωr*).
When an object slips (slides) rather than rolls, its angular velocity is ...........
When an object slips (slides) rather than rolls, its angular velocity is zero.
True or False?
An object rolling without slipping down a slope converts all of its lost gravitational potential energy into translational kinetic energy only.
False.
The gravitational potential energy is transferred to both translational and rotational kinetic energy, since $E_{K t o t a l} = \frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2}$ .
Write the equation for the total kinetic energy of an object rolling down a slope.
$E_{K t o t a l} = \frac{1}{2} m v^{2} + \frac{1}{2} I ω^{2}$
Define the function of a flywheel.
To act as an energy reservoir, storing and supplying rotational kinetic energy when required.
Why is a hoop (wheel)-shaped flywheel preferred over a disc-shaped flywheel of the same mass and radius?
A hoop has a greater moment of inertia ( $I = m r^{2}$ vs $I = \frac{1}{2} m r^{2}$ for a disc), so it stores more rotational kinetic energy.
How does a flywheel smooth out fluctuations in engine torque?
It speeds up or slows down due to its inertia, absorbing sharp fluctuations so the torque delivered is smoother.
Give three ways friction can be reduced in a flywheel system.
- Lubricating the bearings
- Using superconducting bearings so the flywheel levitates
- Operating the flywheel in a vacuum or sealed container
In conventional braking, the kinetic energy store of a vehicle is transferred as waste to the .......... energy store.
In conventional braking, the kinetic energy store of a vehicle is transferred as waste to the thermal energy store.
True or False?
Increasing the mass of a flywheel, while keeping its shape and angular speed constant, decreases the amount of energy it can store.
False.
Increasing the mass increases the moment of inertia, which increases the rotational kinetic energy stored, since $E_{k} \propto I \propto m$ .
State two factors, other than mass and shape, that affect a flywheel's energy storage capacity.
Angular speed and friction.
In a production process such as riveting, how does combining an electric motor with a flywheel benefit the machine?
The motor charges up the flywheel, which then delivers a short burst of energy for the process. This prevents the motor from stalling, so a less powerful motor can be used.

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