Download Friction, Work, and Energy in the Inclined Plane

Friction, Work, and Energy on the Inclined Plane
Objective
To check the validity of Newton’s Second Law by measuring the motion of a cart as it
accelerates up an inclined plane under the action of gravitational and other forces.
Theory
From Newton's second law the force acting on an (inertial) object equals the product
of object's mass and acceleration,
F  m a
F
F
(1)
x
 max
(2)
y
 ma y
(3)
Consider a system of masses m1 placed on a rough,
inclined plane is connected to a string that passes over a
pulley and then is fastened to a hanging m 2 . Since we
can neglect the masses of the string and the pulley, and
the pulley is frictionless the tension at both ends of the
string are the same (magnitude T ). Let us assume m2
accelerates downward with magnitude a. Since the two masses are connected by the
string the acceleration of m1 also has magnitude a.
The equation of motion for the two masses can be written as,
for m 1 :
T  m1g sin   f k  m1a
(4)
N 1  m1 g cos   0
N 1  m1g cos 
(5)
f k  k N 1  k m1g cos 
Then Eq.(4) becomes
T  m1g sin   k m1g cos   m1a
for m 2 :
(6)
m 2 g T  m 2a
(8)
Since
1
(7)
Solving Eq.(7) and Eq.(8), we get
a
m 2 g  m1 g sin   k m1 g cos 
m1  m 2
(9)
Work done by friction:
If an object such as a block is lifted, work is done in order to move the block. The work,
W, done by a force is defined as the component of the force that produces motion parallel
to the direction of the motion (F||) times the displacement of the object on which work is
being done, S, in that same direction:
w F S
(10)
A machine can be defined as any device that multiplies forces or changes the direction of
forces in order to do work. Consider the machine in figure II, which shows a system of
two masses connected by a pulley, where work done on m 2 is used to lift m 1 up the plane.
For the object with a given mass m 2 that moves downward, work is being done on the
object by the force of gravity. The work done is simply the object’s weight times the
distance through which it moved:
w 2  m2 g S
(11)
Since m 1 and m 2 are connected by ropes, then the vertical distance S that m 2 moves
downward is the same distance along the path of the inclined plane that m 1 moves. The
vertical distance that m 1 moves up the plane is related to this distance by
sin  
y 2  y1
S
The total work done on m 1 in order to move it is due to two distinctive kinds of forces:
conservative and non-conservative. A force is conservative when it does no work on an
object that moves around a closed path (the object starts and finishes at the same point).
The gravitational force is a conservative force; hence, any work done by or against
gravity within the system of two blocks is conservative work.
w g  (m1 g sin  )S
(12)
The second component of work in our system is due to non-conservative forces. A force
is non-conservative (or dissipative) if the work it does on an object moving between two
points depends on the path of the motion between the points. Useful work is always lost
to the kinetic frictional force because it dissipates into heat, which is un-recoverable in
our system to do useful work. The non-conservative work for this system is then defined
2
by the frictional force times the distance through which the block moves. The expression
for the work lost due to friction, w nc is:,
w nc  f k S = (k m1 g cos  )S
(13)
Apparatus:
A wooden block, an inclined plane with pulley, cord, weights, a balance, and a meter
stick.
Procedure:
1) Place the wooden block at the bottom of the inclined plane [angle of
inclination = 40]
2) Determine the time, t , that the wooden block will travel distance, S.
3) Repeat paragraph 2 many times for the same distance S.
1
2
4) Find the average time, t , and calculate the acceleration of the block S  at 2 .
5) Coefficient
of
kinetic
friction
is
given
m 2 g  m1 g sin   (m1  m 2 )a
k 
.
m1 g cos 
6) Use eq. (7) or (8) to evaluate the tension in the rope.
7) Use eqs. (11- 13) work done within the system of two blocks.
3
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