91Ó°ÊÓ

The given data can be listed below,

• The number of colas is, N = 12.
• The force on the cola is, F = 36 N.
• The distance travelled by dog is, s = 1.20 m.

The frictional force is defined as the force operating in the opposite direction between two surfaces. It is not a force of conservatism.

The free body diagram of the system is given by,

The work done without frictional force is given by,

$$\begin{gathered} W_{tot}=\stackrel{→}{F}.\stackrel{→}{s} \\ =Fs\cos \mathrm{ϕ} \end{gathered}$$

Here, F is the force on the cord, and s is the displacement.

The total work done on the system is given by,

$$\begin{gathered} W_{tot}=K_2-K_1 \\ =\frac{1}{2}m{v}_2^2 \end{gathered}$$

On comparing two equations, the final velocity of the pack is given by,

$$\begin{gathered} Fs=\frac{1}{2}m{v}_2^2 \\ {v}_2=\sqrt{\frac{2Fs}{m}} \end{gathered}$$

Substitute all the values in the above,

$$" width="9" height="19" role="math">v2=Fsm=2×36N1.20m4.30kg=20.09m/s2=4.48m/s

Thus, the final velocity of the pack without friction is 4.48 m/s.

The work done on the pack when the kinetic friction is acting on the system is given by,

$$\begin{gathered} W_{tot}=Fs-f_{k}s \\ =Fs-{\mathrm{μ}}_{k}mgs \end{gathered}$$

Also, the total work done is given by,

$$\begin{gathered} W_{tot}=\frac{1}{2}m{v}_2^2-\frac{1}{2}m{v}_1^2 \\ =\frac{1}{2}m{v}_2^2-0 \end{gathered}$$

Comparing both the equations, the velocity of the pack when friction is applied is given by,

role="math" localid="1664892476366" $$\begin{gathered} Fs-{\mathrm{μ}}_{k}mgs=\frac{1}{2}m{v}_2^2 \\ {v}_2^2=\frac{2(Fs-{\mathrm{μ}}_{k}mgs)}{m} \\ {v}_2 \\ =\sqrt{\frac{2(Fs-\mathrm{μ}\_kmgs)}{m}} \end{gathered}$$

Here, F is the force acting on the pack,${\mathrm{μ}}_k$ is the coefficient of kinetic friction, and m is the mass of the pack.

Substitute all the values in the above,

$$\begin{gathered} v_2=\sqrt{\frac{2\left(Fs-{\mathrm{μ}}_{k}mgs\right)}{m}} \\ =\sqrt{\frac{2\left(36N×1.2m-0.30kg\left(9.8m/s^2\right)1.2m\right)}{4.3kg}} \\ =\sqrt{13.03}m/s \\ =3.61m/s \end{gathered}$$

Thus, the final velocity of the pack when friction is present is 3.61 m/s.