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Colt1911 [192]
1 year ago
15

A child of mass m is at the edge of a merry-go-round of diameter d. When the merry-go-round is rotating with angular acceleratio

n α, the torque on the child is τ. The child moves to a position half way between the center and edge of the merry-go-round, and the angular acceleration increases to 2α. The torque on the child is now
Physics
1 answer:
dem82 [27]1 year ago
7 0

Answer:

The torque on the child is now the same, τ.

Explanation:

  • It can be showed that the external torque applied by a net force on a rigid body, is equal to the product of the moment of inertia of the body with respect to the axis of rotation, times the angular acceleration.
  • In this case, as the movement of the child doesn't create an external torque, the torque must remain the same.
  • The moment of inertia is the sum of the moment of inertia of the merry-go-round (the same that for a solid disk) plus the product of  the mass of the child times the square of the distance to the center.
  • When the child is standing at the edge of the merry-go-round, the moment of inertia is as follows:

       I_{to} = I_{d} + m*r^{2}  = m*\frac{r^{2}}{2} +  m*r^{2} = \frac{3}{2}*  m*r^{2} (1)

  • So, τ = 3/2*m*r²*α (2)
  • When the child moves to a position half way between the center and the edge of the merry-go-round, the moment of inertia of the child decreases, as the distance to the center is less than before, as follows:

       I_{t} = I_{d} + m*\frac{r^{2}}{4}   = m*\frac{r^{2}}{2} + m*\frac{r^{2}}{4}  = \frac{3}{4}*  m*r^{2} (3)

  • Since the angular acceleration increases from α to 2*α, we can write the torque expression as follows:

       τ = 3/4*m*r² * (2α) = 3/2*m*r²

        same result than in (2), so the torque remains the same.

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