Question

a) State the principle of work and energy
b) If the motor exerts a constant force of 400 N on the cable, determine the speed of the 50-
kg crate when it travels s = 10 m up the plane, starting from rest. The coefficient of kinetic
friction between the crate and the plane is µk = 1/2.

Answer 1

Answer:

Let’s start by looking at the net work done on a particle as it moves over an infinitesimal displacement, which is the dot product of the net force and the displacement:

\[d{W}_{\text{net}}={\overset{\to }{F}}_{\text{net}}·d\overset{\to }{r}.\]

Newton’s second law tells us that

\[{\overset{\to }{F}}_{\text{net}}=m(d\overset{\to }{v}\text{/}dt),\]

, so

\[d{W}_{\text{net}}=m(d\overset{\to }{v}\text{/}dt)·d\overset{\to }{r}.\]

For the mathematical functions describing the motion of a physical particle, we can rearrange the differentials dt, etc., as algebraic quantities in this expression, that is,

\[d{W}_{\text{net}}=m(\frac{d\overset{\to }{v}}{dt})·d\overset{\to }{r}=md\overset{\to }{v}·(\frac{d\overset{\to }{r}}{dt})=m\overset{\to }{v}·d\overset{\to }{v},\]