Page 17 - Mechanical Engineers Reference Book

1/6 Mechanical engineering principles
Angular kinetic energy about an axis 0 is given by 1hIow2.
Work done due to a torque is the product of torque by
angular distance and is given by TO.
Power due to torque is the rate of angular work with respect
to time and is given by Td0ldt = Tw.
Friction: Whenever two surfaces, which remain in contact,
move one relative to the other there is a force which acts
tangentially to the surfaces so as to oppose motion. This is
known as the force of friction. The magnitude of this force is
pR, where R is the normal reaction and p is a constant known
as the coefficient of friction. The coefficient of friction de-
pends on the nature of the surfaces in contact.

1.3.2 Linear and angular motion in two dimensions
Constant acceleration: If the accleration is integrated twice and
the relevant initial conditions are used, then the following
equations hold:
Linear motion Angular motion
Figure 1.2 x = vlt + ;a? 0 = w1t + iff?
v2 = v, + at w2 = w1 + at
Both angular velocity and accleration are related to linear
=
motion by the equations v = wx and a = LYX (see Figure 1.2). vt = v: + 2ax 4 w: + 2a8
Torque (T) is the moment of force about the axis of
rotation: Variable acceleration: If the acceleration is a function of
time then the area under the acceleration time curve repre-
T = IOU sents the change in velocity. If the acceleration is a function of
A torque may also be equal to a couple, which is two forces displacement then the area under the acceleration distance
equal in magnitude acting some distance apart in opposite curve represents half the difference of the square of the
velocities (see Figure 1.4).
directions. Curvilinear motion is when both linear and angular motions
Parallel axis theorem: if IGG is the moment of inertia of a
body of mass m about its centre of gravity,, then the moment of are present.
If a particle has a velocity v and an acceleration a then its
inertia (I) about some other axis parallel to the original axis is motion may be described in the following ways:
given by I = IGG + m?, where r is the perpendicular distance
between the parallel axes. 1. Cartesian components which represent the velocity and
Perpendicular axis theorem. If Ixx, Iyy and Izz represent acceleration along two mutually perpendicular axes x and
the moments of inertia about three mutually perpendicular y (see Figure 1.5(a)):
axes x, y and z for a plane figure in the xy plane (see Figure
1.3) then Izz = Ixx + Iyy.
Angular momentum (Ho) of a body about a point 0 is the a a
moment of the linear momentum about that point and is wZOo.
The angular momentum of a system remains constant unless 2
acted on by an external torque.
Angular impulse is the produce of torque by time, i.e.
angular impulse = Tt = Icy . t = I(w2 - q), the change in
angular momentum.

Y

t X
dv a= *. dv
a = - oradt=dv
dt dt dx
dv
Area a.dt a=v -
dx
= vz - v, or adx = vdv

0 X
Figure 1.3 Figure 1.4