Rotational Dynamics Flashcards
An unbalanced torque causes a change in the
angular rotational motion of an object
definition of moment of inertia
moment of inertia of an object as a measure of its resistance to angular acceleration about a given axis
moment of inertia depends on
mass and the distribution of mass about a given axis of rotation
I = mr^2
point mass
I = moment of inertia (kg m^2) m = mass r = distance from centre of axis of rotation to point mass
Moment of inertia = I
measured in kg m^2
I = 1/12ml^2
rod about a centre
where the centre of the rod is at the centre of the axis of rotation
I = moment of inertia (kg m^2) m = mass (kg) l = length of rod (m)
I = 1/3ml^2
rod about end
where the end of the rod is at the centre of the axis of rotation
I = moment of inertia (kg m^2) m = mass (kg) l = length of rod (m)
I = 1/2mr^2
disc about centre
a 3d disc, unlike the point mass which is flat.
I = moment of inertia (kg m^2) m = mass (kg) r = distance from end of disc to centre of axis of rotation
I = 2/5mr^2
sphere about centre
I = moment of inertia (kg m^2) m = mass (kg) r = radius of the sphere, distance from centre of axis of rotation to edge of sphere
Torque
T
(N m)
Torque
T = Fr
T = Torque (N m) F = Force (N) r = perpendicular distance between direction of force and axis of rotation (m)
If the force is not applied perpendicular i.e an angle of 15 degrees then its perpendicular component would have to be calculated by
T = Fr T = sin15 x r T = 0.65r
An unbalanced Torque will produce an angular acceleration about an axis of rotation
Tᵤₙ = Iα
Tᵤₙ = Unbalanced Torque (N m) I = Moment of inerti (kg m^2) α = angular acceleration (rad s-2)
Angular Momentum
L
kg m^2 s-1
L = mvr
L= mr^2w
L=Iw
L = angular momentum (kg m^2 s-1) m = mass (kg) v = linear velocity (ms-1) r = radius (m) I = moment of inertia (kg m^2) w = angular velocity (rad s-1)
The principle of conservation of angular momentum
Total angular momentum before = Total angular momentum after
The conservation of angular momentum formula
Not in data book!
I₁w₁= I₂w₂
Just as linear momentum is conserved when two or more objects collide in absence of external forces
I = moment of inertia (kg m^2) w = angular velocity (rad s-1)
when combining different objects use this formula
I = Σmr^2
The sum of moment of inertia , add them together
Moment of inertia = the sum of the mr^2
find their I then add them together
Rotational Kinetic Energy
Ek = 1/2Iw^2
Ek + kinetic energy of a rotating body (J)
I = moment of inertia (kg m^2)
w = angular velocity (kg m^2)
Ep = Ek (linear) + Ek (rotational)
so
mgh = 1/2mv^2 +1/2lw^2
worked example of rotational kinetic energy
Ep lost = Ek gained mgh = 1/2mv^2 + 1/2lw^2 since w = v/r mgh = 1/2mv^2 + 1/2l x (v^2/r^2) take v^2 out both leaving as a fraction (1/r^2) mgh = v^2 (1/2m + 1/2l x (1/r^2)
this is now solvable with numerical values.
Ep lost =
Ek gained
Why does an object no longer travel in a circular path when the speed is reduced
Tension will be reduced
Weight is bigger than the centripetal force
An object at its highest point about its rotation of axis will have the equation
Fᵤₙ = Tension + Weight
Both weight and tension are in the downward direction towards the centre of the axis
