91Ó°ÊÓ

Mass of the crate m = 300 kg

Velocity of the crate v = 1.20 m/s

Coefficient of friction$\mathrm{μ}=0.400$

The problem is based on the principle of conservation of mechanical energy. According to this principle the energy can neither be created nor be destroyed; it can only be internally converted from one form to another if the forces doing work on the system are conservative in nature. By using the concept of frictional force and mechanical energy of non - conservative systems we can find the required values.

Formula:

$K=\frac{1}{2}×m{v}^2$

role="math" localid="1661403272907" $$\begin{gathered} f_k={\mathrm{μ}}_{k}N \\ {E}_{mec}=K+W_{friction} \\ {W}_{friction}=f_{k}d \end{gathered}$$

Kinetic Energy supplied to the crate

$K=\frac{1}{2}×m{v}^2$

$⇒K=\frac{1}{2}×300×{\left(1.20\right)}^2$

$⇒K=\frac{432}{2}$

$⇒K=216J$

Determine the frictional force experienced on the particle as:

$$\begin{gathered} f_k={\mathrm{μ}}_k\mathrm{N} \\ ⇒{\mathrm{f}}_{\mathrm{k}}=0.4×\mathrm{m}×\mathrm{g} \\ ⇒{\mathrm{f}}_{\mathrm{k}}=0.4×300×9.8 \\ {\mathrm{f}}_{\mathrm{k}}=1176\mathrm{N} \\ ≈1.18×{10}^3\mathrm{N} \end{gathered}$$

$E=KineticEnegry+workdoneagainstthefriction$

$WorkDoneagainstthefriction=W_f=f_{k}d$

To find value of d,

Acceleration of the crate can be calculated as,

$$\begin{gathered} f_k=maa \\ =\frac{f_k}{m}a \\ =\frac{1176}{300}a \\ =3.92\frac{m}{s^2} \end{gathered}$$

Time taken by the crate to achieve belts velocity is:

$\begin{array}{l}t=\frac{v}{a}t\\ =\frac{1.20}{3.92}t\\ =0.3061sec\end{array}$

Distance covered by the crate during this time,

$\begin{array}{l}d=v×t\\ =1.20×0.3061\\ =0.3673m\end{array}$

Determine the work done by the friction as:

$⇒W_{friction}=f_{k}d$
$$\begin{gathered} W_{friction}=1176×0.3673W_{friction} \\ =432J \end{gathered}$$

$E_{motor}≠K$

Because some of the energy supplied by the motor is converted into heat energy due to work done against the friction.