M1.6 Types of Motion
M1.6.1
Translational Motion
Assuming constant acceleration = ac
Velocity as a function of time:
v = vo + act
Position as a function of time:
Velocity as a function of position:
s = so + vot + ½ act2
v2 = vo2 + 2ac(s – so)
Example: t = 3.5s, vo = 0, ac = g/2 = 4.9 m/s2
Newton’s Second Law of Motion (translational motion):
F = ma
Translational Kinetic Energy (TKE):
TKE = ½ mv2
6 M1.6.2
Rotational Motion
Assuming constant angular acceleration, αc, about a fixed axis through the centre of gravity
of the body.
Angular velocity as a function of time:
ω = ωo + αct
where, ω and ωo are measured in r / s, and αc in r / s2
ωo is the initial values of the body’s angular velocity
Angular position as a function of time:
θ = θo + ωot + ½ αct2
where, θ and θo are measured in r (radians)
θo is the initial values of the body’s angular position
Velocity as a function of position:
ω2 = ω02 + 2αc(θ – θo)
Newton’s Second Law of Motion (pure rotational motion of a body):
T = IGαc
where, for a pure rotation, IG is the mass moment of inertia for the body about the axis of
rotation through the mass centre of the body.
T is the applied torque about the axis of rotation measured in (N  m) / r (SI)
Rotational Kinetic Energy (RKE):
RKE = ½ IGω2
Note that in general the total kinetic energy of a body consists of the scalar sum of the
body’s translational kinetic energy and rotational kinetic energy about its mass centre. For
simplicity, we will only consider pure rotational systems (i.e. bodies with no translation rotating
about the centre of mass).
7 Mass Moment of Inertia:
The moment of inertia is simply a measurement of an object’s ability to resist changes in
its rotational rate. This figure illustrates the moment of inertia for a cylinder body with respect to
each axis
Example: A flywheel is rotating at 10000 rpm. The flywheel is a flat cylindrical steel disk 60 cm
in diameter and 7.5 cm thick. Assume that the density of steel is 7.86 g/cm3.
1. If it is rotating about the axis of symmetry, what is the rotational kinetic energy?
8 2. Suppose that the flywheel is used as an energy storage device for a vehicle with a total
mass of 900 kg (including the flywheel). If the flywheel is rotating at 10000 rpm, how
many times can the vehicle be accelerated from 0 to 100 km/hr without employing
regenerative breaking?
3. How does the energy storage capacity of the flywheel compare with that of a 12v car
battery with a capacity of 50 amp‐hrs?
4. A liter of gasoline can provide about 3.43x104 kJ of chemical energy. How much energy
is stored in the flywheel measured in terms of volume of gasoline?
5. How long will it take to accelerate the flywheel to 10000 rpm, if you apply a torque of
1 N  m?
9