91Ó°ÊÓ

Given data can be listed below,

• Mass,$\mathrm{M}=7\mathrm{kg}$
• Force,$\mathrm{F}=50 \mathrm{N}$
• The chain straightens out to full length,$L=2.6 m$
• Total distance to pull the end of the chain,$\mathrm{d}=4.5 \mathrm{m}$

The translation work done on the chain is given as,

${\mathrm{W}}_{\mathrm{trans}}=\mathrm{F}\left(\mathrm{d}-\frac{\mathrm{L}}{2}\right)$

Substituting 50 N for F, 4.5 m for d, and 2.6 m for L in the above equation

$$\begin{gathered} {\mathrm{W}}_{\mathrm{trans}}=\left(50\mathrm{N}\right)\left(4.5\mathrm{m}-\frac{2.6\mathrm{m}}{2}\right) \\ =160\mathrm{J} \end{gathered}$$

Part a)

By using conservation of energy, the translational work done is equal to translational kinetic energy, which is written as,

$$\begin{gathered} {\mathrm{W}}_{\mathrm{trans}}={\mathrm{KE}}_{\mathrm{trans}} \\ =\frac{1}{2}{\mathrm{Mv}}^2 \end{gathered}$$

Where v is the speed of the chain relative to the center of the chain.

Substituting 160 J for$W_{trans}$ , and 7 kg for M in the above equation, we get

$$\begin{gathered} \left(160\mathrm{J}\right)=\frac{1}{2}\left(7\mathrm{kg}\right){\mathrm{V}}^2 \\ {\mathrm{v}}^2=\frac{2\left(160\mathrm{J}\right)}{\left(7\mathrm{kg}\right)} \\ \mathrm{v}=6.76\mathrm{m}/\mathrm{s} \end{gathered}$$

Thus, the speed of the chain at the instant is 6.76 m/s.

Part b)

The change in energy is equal to the total work done by the chain is given as,

$$\begin{gathered} ∆E=W_{total} \\ =Fd \end{gathered}$$

Substituting 50 N for F, and 4.5 m for d in the above equation

$$\begin{gathered} ∆\mathrm{E}=\left(50\mathrm{N}\right)\left(4.5\mathrm{m}\right) \\ =225\mathrm{J} \end{gathered}$$

Thus, the change in energy of the chain is 225 J.

Part c)

By using the 1^st law of thermodynamics,

$\mathrm{Δ}E=W_{trans}+\mathrm{Δ}Q$

Where is the change in thermal energy.

Substituting 160 J for$W_{trans}$ , and 225 J for in the above equation

$$\begin{gathered} \mathrm{Δ²Ï}=\mathrm{Δ·¡}-{\mathrm{W}}_{\mathrm{trans}} \\ =225\mathrm{J}-160\mathrm{J} \\ =65\mathrm{J} \end{gathered}$$

Thus, the increase in the thermal energy of the chain is 65 J.