I). mechanical energy is the sum of potential energy and kinetic energy in an object that is used to do work.
ii). the principle of the conservation of mechanical energy states that the mechanical energy in a system remains constant as long as the only forces acting are conservative.
iii). frictional force is a non-conservative force.
which of the following statement below is false?
a) for the motion of a cart on a frictionless incline plane, the magnitude of the change in kinetic energy equals the magnitude of the change in gravitational potential energy
b) for the motion of a cart on an incline plane having a coefficient of kinetic friction of 0.5, the magnitude of the change in kinetic energy equals the magnitude of the change in gravitational potential energy
c) assuming a constant gravitational potential energy for a cart, the work-done by a net force on the cart equals the change in its kinetic energy
d) in an ideal free fall, the magnitude of the change in gravitational potential energy equals the magnitude of the change in gravitational potential energy

false statement : b )  For the motion of a cart on an incline plane having a coefficient of kinetic friction of 0.5, the magnitude of the change in kinetic energy equals the magnitude of the change in gravitational potential energy

Explanation:

mechanical energy = potential energy + kinetic energy = constant

differentiating both side

Δ potential energy + Δ kinetic energy = 0

Δ potential energy = -  Δ kinetic energy

first statement is true.

Friction is a non conservative force so inter-conversion of potential and kinetic energy is not possible in that case. In case of second  option,  the correct relation is as follows

change in gravitational potential energy = change in kinetic energy + work done against friction .

So given 2 nd  option is incorrect.

In case of no change in gravitational energy , work done is equal to

change in kinetic energy.

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there are no contact forces mentioned on the list of choices that you've provided.