Flashcards in Unit 5: Rotational dynamics Deck (58):
What are angles and angular displacements measured in?
What is the definition of a radian?
When the arc length is equal to the radius, the angle θ is equal to one radian
What is the formula for angular velocity?
ω = Δθ/Δt rad(s^-1)
What is the formula for velocity in terms of angular velocity?
v = rω
When does angular acceleration arise?
When a particle increases its speed of rotation
What is angular acceleration defined as?
The rate of change of angular velocity
What is the formula for angular acceleration?
α = Δω/Δt rad(s^-2)
What must an increase in angular acceleration be accompanied by?
A corresponding increase in the tangential acceleration
What is the formula for tangential acceleration in terms of angular acceleration?
a = rα
What do the dynamics involving movement about a circle assume?
It considers a body that can be treated as a single particle where all the mass is centred at a point but this simplistic assumption cannot be made in more realistic situations
What does the dynamics of a rotating body depend upon?
The distribution of the mass of the body in relation to a particular axis of rotation
What is a body composed of?
An infinite number of point masses (or elemental particles) that move collectively - each point mass within the body will have a different radius from the axis of rotation and will therefore rotate with a different tangential velocity
What does inertia mean?
The resistance of an object to a change in its motion
What is the formula for the moment of inertia?
I = Σm(r^2)
What are the units of the moment of inertia?
When is the summation for the moment of inertia replaced by integration?
In the case of a rigid body with a mass that is continuous
What affects the value of the moment of inertia?
The precise location of the axis of rotation
What increases the moment of inertia?
Distributing the mass further from the centre of rotation
What is the moment of inertia of a hoop about the central axis?
I = M(R^2)
What is the moment of inertia of a solid sphere about any diameter?
I = 0.4M(R^2)
What is the moment of inertia of a solid cylinder/disc about the central axis?
I = 0.5M(R^2)
What happens to the moments of inertia of two objects that share a common axis of rotation?
Their individual moments of inertia are added together to give a moment of inertia for the combined system
What is a flywheel?
A mechanical device with a large moment of inertia that stores rotational energy
What is the formula for rotational kinetic energy?
Ek = 0.5I(ω^2)
When a body is rotating about a centre of rotation, what does it possess energy due to?
Its rotational movement
What are the four rotational equations of motion?
ω2 = ω1 + αt, θ = ω1t + 0.5α(t^2), (ω2)^2 = (ω1)^2 + 2αθ and θ = 0.5(ω1 + ω2)/t
What is torque?
The moment (or turning effect) of a force about a point
What is the definition of torque?
A torque is the product of a force and the perpendicular distance of the line of action of the force from a perpendicular point of rotation. The unit of torque is newton metre (Nm). The convention adopted is that positive torques produce a clockwise rotation
What is the formula for torque in terms of force?
T = Fs
What does a couple consist of?
Two equal, opposite and parallel forces
Where is the axis of rotation for a couple?
The axis of rotation is at the centre of an object or system
Do the lines of action of the forces coincide in a couple?
What is the torque produced by a couple equal to?
The product of one of the forces and the perpendicular distance between the forces
What are the effects of couples?
Couples produce rotational movement and an angular acceleration
What is the formula for torque in terms of the moment of inertia?
T = Iα
What is angular momentum?
The angular momentum of a particle about an axis is the product of its linear momentum and the perpendicular distance of the particle from the axis
What is the formula for angular momentum in terms of m, v and r?
L = mvr
What is the formula for angular momentum in terms of m, ω and r?
L = mω(r^2)
What is the formula for angular momentum in terms of the moment of inertia and when does this formula apply?
L = Iω and this only applies to a rigid body with a fixed axis of rotation
Is angular momentum a vector quantity?
In which direction does angular momentum act?
Angular momentum has an associated direction which is along the axis of rotation and hence perpendicular to the plane of rotation
What can be said about angular momentum by convention?
The direction of the angular momentum vector is towards an observer if the direction of rotation is anticlockwise
What does the principle of the conservation of angular momentum state?
The total angular momentum at some initial time is equal to the total angular momentum at some time later provided no external torque acts upon the system, no matter what takes place within the system
What may the conservation of angular momentum result in?
Bodies rotating in opposite directions as a result of a collision
How does the conservation of angular momentum apply to the elliptical motion of planets orbiting the sun?
The direction of the gravitational force is perpendicular to the orbital motion and hence the force exerts no torque on the planet
If the angular momentum of a planet is conserved, what must happen if the moment of inertia decreases and what verification is there for this?
The angular velocity of the planet must increase and this has been verified since planets move faster when closer to the sun and slower when further away
What is the initial setup for the conservation of angular momentum experiment involving a gyroscope?
A spinning gyroscope is fixed in a horizontal position on a vertical pole attached to a turntable that is free to rotate, the gyroscope will continue spinning in its fixed position and the turntable will remain still
What happens if the spinning gyroscope is now positioned vertically?
The turntable will rotate in the opposite direction to the spinning gyroscope so that the angular momentum is conserved
What happens in the case for a spinning gyroscope that is not fixed but is on a vertical pivot?
The action of gravity forces the gyroscope to rotate slowly about the pivot point i.e. it precesses in an anticlockwise direction
What is angular impulse?
The change in total angular momentum
What units does angular impulse have?
The same as angular momentum
What is torque equal to?
The rate of change of angular momentum
What is the formula for torque in terms of angular momentum?
T = ΔL/Δt
What is work done the product of?
The force and the distance moved
What is the formula for work done in terms of torque?
W = Tθ
What can the work done be used for?
It can be used to overcome any resistive forces that may be present in the rotating system such as friction or it may be used to increase the rotational kinetic energy of the rigid body
What is power?
The rate of doing work