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Title: Rolling on an inclined plane
Aim: To find
the radius of gyration of a body rolling on an inclined plane.
Apparatus:
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THEORY:
*GENERAL
KNOWLEDGE*
An inclined plane can be defined as
any plane surface positioned at an angle with respect to the horizontal plane.
Geometrically, it can also be called right-angled triangle in which, the
hypotenuse is the inclined plane, the base is the horizontal plane and the
perpendicular is the vertical height. For experimental purposes, the inclined
plane consists of two plane parallel plates or surfaces hinged together so that
they can be rotated about horizontal axis as the horizontal and inclined
planes. The inclined plane is smoothened to minimize the effects of friction
and is also provided with a pulley system for different motion related
experiments.
Theory 1:
The inclined plane reduces the effect of the force of gravity
on an object on the inclined surface by changing the angle of application of
the force component.
As per Newton’s second law of motion, Force acting on a body
can be described as,
F = Ma Where F = Vector sum
of all the forces (or net resultant force) acting on the object, m = mass of
the object and a = acceleration produced in the body due to the effect of
forces.
Suppose, an object of mass m1 is placed on an
inclined plane. It can move upward, downward or remain stationary. In all of
these different situations, net force acting on the object will be different.
The situations are analyzed using “Free
Body Diagrams (FBD)”.
a.)
Without
Friction
In diagram 2, consider the object of mass m1
placed on inclined plane. Various forces acting on the object are
(i)
Force of gravity
acting vertically downward
(ii)
Tension in the
string due to mass m2 suspended from the string connecting the two
objects
The free body diagram of the objects under the effect of
these forces is as shown in diagram 3. In the diagram,
T = tension in the cord
m1 = mass of the object placed on the inclined
plane (i.e., roller)
m2 = mass of the object hanging from the pulley
(i.e., weight pan with weights)
N = Normal reaction force due to weight component of the
object
q = Angle of inclination
g = Acceleration due to gravity
ax = Acceleration along x-axis
ay = Acceleration along y-axis
b.)
With
Friction
In this case, various forces acting on the object are,
(i)
Force of gravity
acting vertically downward.
(ii)
Tension in the
string due to mass m2 suspended from the string connecting the two
objects.
(iii)
Force of friction
on the roller.
The free body diagram of the objects under the effect of
these forces is as shown in diagram 4.
This case is similar to the previous one except that force of
friction between inclined surface and roller is acting on the roller so as to
oppose the motion of the roller. The frictional force may be dynamic (acting on
the body in motion) or static (acting on the body, when it is stationary and
resists the tendency of the body to move from stationary position). The
direction of frictional force is opposite to the direction of motion. When the
body moves upward on the inclined surface, the frictional force (f) acts in the
downward direction, as shown in the diagram 4. If the body is rolling down the
inclined surface, frictional force acts in the upward direction.
If, ‘f’ is the force of friction, m is the co-efficient of friction between
roller and inclined surface, then
……….
(5)
In case of static friction, force of static friction (fs,
that opposes the movement of a body from the resting position) can be described
as
Where, fs is the force applied to just start the
motion of body and ms
is coefficient of static friction between the two bodies. When the bodies are
in motion, let fk be the force required to maintain the uniform
speed of motion and mk
be coefficient of static friction, then
For
Procedures, See your Mechanical Laboratory Manual.
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Observations
:-
1.
Applications:-
PRECAUTIONS:
For Precautions, See General Laboratory Precautions
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