A rectangular board is made of two pieces as shown. The shaded piece and the uns

Question

A rectangular board is made of two pieces as shown. The shaded piece and the unshaded piece are made of materials of different uniform mass densities. The board is placed on a frictionless pivot. It remains at rest as shown 1. Frictionless pivot a. Is the mass of the unshaded piece (m) greater than, less than, or Unshaded piece Shaded piece equal to the mass of the shaded piece (n)? Explain. b. Mark the approximate location of the center of mass of the board with an "x" Explain how you determined the location. If you do not have enough information, state so explicitly

Explanation / Answer

Since the rectangular board is in static equilibrium, the net torque about any point is zero.
Consider the mass of unshaded piece as Mu and its center of mass from the pivot is Ru. Also, consider the mass of the shaded piece is Ms and its center of mass from the pivot is Rs.
Consider the net torque about the pivot,
Mu Ru - Ms Rs = 0
Mu Ru = Ms Rs
Ru / Rs = Ms / Mu              ...(1)

From the figure, it can be understood that Ru > Rs
or, Ru / Rs > 1                   ...(2)

Combining (1) and (2),
Ms / Mu > 1
ie, Ms > Mu
So mass of shaded piece is greater.

b)
Since the board is at rest, the center of mass must coincide with the position of the pivot. Other wise, there will be a net torque and the board will not balance.

2)
Consider that each block has mass m. In the first figure, there are two blocks with mass m on the left and 4 blocks on the right.
The center of mass of the blocks on the left from the 50 cm mark = 45 cm
The center of mass of the blocks on the right from the 50 cm mark = 22.25 cm
Torque on the left due to the two masses = 2m g x 45 = 90 mg N. cm
Torque on the right due to 4 masses = 4m g x 22.5 = 90 mg N. cm

Since the two torques are equal, the meter stick will balance.

b)
Similar to the above part, torque on the left = 2mg x 25 = 50 mg N. cm
Torque on the right = mg x 10 + mg x 45 = 55 mg N cm
Since the torques are not equal, the meter stick will not balance.

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