Problem 2: You have been hired to design a family-friendly see-saw. Your design will feature a uniform board of mass \( M \) and length \( L \) that can be moved so that the fulcrum (pivot) is a distance \( d \) from the center of the board. This will allow riders to achieve static equilibrium even if they are of different masses, which is typical. You have decided that each rider will be positioned so that his/her center of mass will be a distance \( x_{\text {offset }} \) from the end of the board when seated, as shown. A child, seated on the right, has mass \( m \), and an adult, seated on the left, has a mass that is a multiple \( n \) of the mass of the child. Calculate all torques relative to the position of the fulcrum, and treat counterclockwise torques as positive.
Part (a) Enter an expression for the torque due to the adult rider, who is seated on the left. Expression :
\[
\tau_{\text {adult }}=
\]
\( \_\_\_\_ \)
Select from the variables below to write your expression. Note that all variables may not be required.
\[
\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\theta}, \mathbf{d}, \mathbf{g}, \mathbf{h}, \mathbf{j}, \mathbf{k}, \mathbf{L}, \mathbf{m}, \mathbf{M}, \mathbf{n}, \mathbf{P}, \mathbf{t}, \mathbf{x}_{\text {offset }}
\]
Part (b) Enter an expression for the torque due to the child rider, who is seated on the right.
Expression :
\[
\tau_{\text {child }}=
\]
\( \_\_\_\_ \)
Select from the variables below to write your expression. Note that all variables may not be required.
\[
\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\theta}, \mathbf{d}, \mathbf{g}, \mathbf{h}, \mathbf{j}, \mathbf{k}, \mathbf{L}, \mathbf{m}, \mathbf{M}, \mathbf{n}, \mathbf{P}, \mathbf{t}, \mathbf{x}_{\text {offset }}
\]
Part (c) Enter an expression for the torque on the board due to its weight.
Expression :
\[
\tau_{\text {board }}=
\]
\( \_\_\_\_ \)
Select from the variables below to write your expression. Note that all variables may not be required.
\[
\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\theta}, \mathbf{d}, \mathbf{g}, \mathbf{h}, \mathbf{j}, \mathbf{k}, \mathbf{L}, \mathbf{m}, \mathbf{M}, \mathbf{n}, \mathbf{P}, \mathbf{t}, \mathbf{x}_{\text {offset }}
\]
Part (d) Determine the distance, \( d \), that achieves equilibrium in terms of \( n, g \), and the masses and lengths in the problem.
Expression :
\[
d=
\]
\( \_\_\_\_ \)
Select from the variables below to write your expression. Note that all variables may not be required.
\[
\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\theta}, \mathbf{d}, \mathbf{g}, \mathbf{h}, \mathbf{j}, \mathbf{k}, \mathbf{L}, \mathbf{m}, \mathbf{M}, \mathbf{n}, \mathbf{P}, \mathbf{t}, \mathbf{x}_{\text {offset }}
\]
Part (e) Determine the magnitude of the force exerted on the fulcrum (pivot) of the see-saw.
Expression :
\[
F_{\mathrm{net}}=
\]
\( \_\_\_\_ \)
Select from the variables below to write your expression. Note that all variables may not be required.
\[
\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\theta}, \mathbf{d}, \mathbf{g}, \mathbf{h}, \mathbf{j}, \mathbf{k}, \mathbf{L}, \mathbf{m}, \mathbf{M}, \mathbf{n}, \mathbf{P}, \mathbf{t}, \mathbf{x}_{\text {offset }}
\]
