Chapter 5, Problem 138P

(a)

To determine

The center of mass of the you-raft system from the end of the pier when you are at the very rear.

Answer to Problem 138P

The center of mass of the you-raft system from the end of the pier when you are at the very rear is at $5.5\text{ m}$ .

Explanation of Solution

Given:

Length and mass of the raft are $6\text{ m}$ and $120\text{ kg}$ respectively.

Your mass is $60\text{ kg}$ .

Formula used:

The edge of the pier is taken as the origin and the right side of the pier is taken as positive $X-$ direction, as shown in the Figure $1$ .

  

Figure: $1$

In the figure $2$ , $d$ is the distance of the end of the raft from the pier after you have walked to its front.

  

Figure: $2$

The center of mass of the you-raft system ( $x_{cm}$ ) before you walk to the front of the raft is given by

  $x_{cm}=\frac{m_y{x}_{y,i}+m_r{x}_{r\_cm,i}}{m_y+m_r}$ $\left(1\right)$

Where, $m_y$ is your mass

  $m_r$ is mass of the raft

  $x_{y,i}$ is your initial position

  $x_{r\_cm,i}$ is the initial position of the center of mass of the raft, that is before you have walked to the front of the raft

Calculation:

By substituting numerical values in equation $\left(1\right)$ ,

  $x_{cm}=\frac{(60\text{ kg})(7.5\text{ m})+(120\text{ kg})(4.5\text{ m})}{60\text{ kg}+120\text{ kg}}$

  $=5.5\text{ m}$ .

Conclusion:

The center of mass of the you-raft system from the end of the pier when you are at the very rear is at $5.5\text{ m}$ .

(b)

To determine

The center of mass of the you-raft system from the end of the pier when you are at front

Answer to Problem 138P

The center of mass of the you-raft system from the end of the pier when you are at front is at $2.5\text{ m}$ .

Explanation of Solution

Formula used:

  $x_{cm}$ before you walk to the front of the raft is given by

  $x_{cm}=\frac{m_y{x}_{y,i}+m_r{x}_{r\_cm,i}}{m_y+m_r}$ $\left(2\right)$

  $x_{cm}$ after you have walked to the front of the raft is given by

  $x_{cm}=\frac{m_y{x}_{y,f}+m_r{x}_{r\_cm,f}}{m_y+m_r}$ $\left(3\right)$

Where, $m_y$ is your mass

  $m_r$ is mass of the raft

  $x_{y,i}$ is your initial position

  $x_{r\_cm,i}$ is the initial position of the center of mass of the raft, that is before you have walked to the front of the raft

  $x_{y,f}$ is your final position

  $x_{r\_cm,f}$ is the final position of the center of mass of the raft, that is after you have walked to the front of the raft

Calculation:

The distance of the raft from the pier after you have walked to the front of the raft is,

  $d=0.50\text{ m}+x_{y,f}$ $\left(4\right)$

Because $F_{ext,net}=0$ , the center of mass remains fixed and hence one can equate the expressions $\left(2\right)$ and $\left(3\right)$ , that is

  $m_y{x}_{y,i}+m_r{x}_{r\_cm,i}=m_y{x}_{y,f}+m_r{x}_{r\_cm,f}$

  $x_{y,f}=x_{y,i}-\frac{m_r}{m_y}\left(x_{r\_cm,f}-x_{r\_cm,i}\right)$

Since, $x_{r\_cm,f}-x_{r\_cm,i}$ = $x_{y,f}$

Substituting $x_{y,f}$ for $x_{r\_cm,f}-x_{r\_cm,i}$ in the above equation,

  $x_{y,f}=\frac{m_y{x}_{y,i}}{m_y+m_r}$

By substituting the numerical values in the above equation,

  $x_{y,f}=\frac{(60\text{ kg})(6\text{ m})}{60\text{ kg}+120\text{ kg}}$ = $2\text{ m}$

Substituting this value in equation (4),

  $d=2\text{ m}+0.5\text{ m}=2.5\text{ m}$

Conclusion:

The center of mass of the you-raft system from the end of the pier when you are at front is at $2.5\text{ m}$ .

(c)

To determine

The distance between you and the end of the pier when you are at the front of the raft.

Answer to Problem 138P

The distance between you and the end of the pier when you are at the front of the raft is at $4.5\text{ m}$ .

Explanation of Solution

Formula used:

  $x_{cm}$ after you’ve walked to the front of the raft is given by

  $x_{cm}=\frac{m_{y}d+m_r(d+x_{r\_cm})}{m_y+m_r}$ $\left(5\right)$

Where, $m_y$ is your mass

  $m_r$ is mass of the raft

  $x_{y,i}$ is your initial position

  $x_{r\_cm}$ is the center of mass of the raft

  $d$ is the distance of the raft from the pier after you have walked to the front of the raft

Calculation:

Substituting numerical values in equation in equation $\left(5\right)$ ,

  $x_{cm}=\frac{(60\text{ kg})(2.5\text{ m})+(120\text{ kg})(2.5\text{ m}+3\text{ m})}{60\text{ kg}+120\text{ kg}}$

  $=4.5\text{ m}$

Conclusion:

The distance between you and the end of the pier when you are at the front of the raft is at $4.5\text{ m}$ .