Page 139 - Class-11-Physics-Part-1_Neat

WORK, ENERGY AND POWER 125

4
= 1.25 × 10 J
We obtain
x = 2.00 m
m
We note that we have idealised the situation.
The spring is considered to be massless. The
surface has been considered to possess
negligible friction. t
We conclude this section by making a few Fig. 6.9 The forces acting on the car.
remarks on conservative forces.
(i) Information on time is absent from the above 1 2
discussions. In the example considered ∆K = K − K 0= − 2 m v
i
f
above, we can calculate the compression, but The work done by the net force is
not the time over which the compression 1
−
=
−
2
occurs. A solution of Newton’s Second Law W kx m g x m
µ
m
for this system is required for temporal 2
information. Equating we have
(ii) Not all forces are conservative. Friction, for
1 2 1 2
example, is a non-conservative force. The m v = k x m + m g x m
µ
2 2
principle of conservation of energy will have 3 3
to be modified in this case. This is illustrated Now µmg = 0.5 × 10 × 10 = 5 × 10 N (taking
-2
g =10.0 m s ). After rearranging the above
in Example 6.9.
equation we obtain the following quadratic
(iii) The zero of the potential energy is arbitrary.
equation in the unknown x .
It is set according to convenience. For the m
spring force we took V(x) = 0, at x = 0, i.e. the 2 + − m v =
2
k x m 2 m g x m 0
µ
unstretched spring had zero potential
energy. For the constant gravitational force
mg, we took V = 0 on the earth’s surface. In
a later chapter we shall see that for the force
due to the universal law of gravitation, the where we take the positive square root since
zero is best defined at an infinite distance x is positive. Putting in numerical values we
m
from the gravitational source. However, once obtain
the zero of the potential energy is fixed in a x = 1.35 m
m
given discussion, it must be consistently which, as expected, is less than the result in
adhered to throughout the discussion. You Example 6.8.
cannot change horses in midstream ! If the two forces on the body consist of a
conservative force F and a non-conservative
c
Example 6.9 Consider Example 6.8 taking force F , the conservation of mechanical energy
nc
t
the coefficient of friction, µ, to be 0.5 and formula will have to be modified. By the WE
calculate the maximum compression of the theorem
spring. (F + F ) ∆x = ∆K
nc
c
But F ∆x = − ∆V
c
Answer In presence of friction, both the spring Hence, ∆(K + V) = F ∆x
nc
force and the frictional force act so as to oppose ∆E = F ∆x
nc
the compression of the spring as shown in where E is the total mechanical energy. Over
Fig. 6.9. the path this assumes the form
We invoke the work-energy theorem, rather E −− −− − E = W
f i nc
than the conservation of mechanical energy.
where W nc is the total work done by the
The change in kinetic energy is non-conservative forces over the path. Note that

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