A 1.2kg mass is located at the origin of an xy coordinate system, a 2.5 kg mass is at x = 2m, y = 0, and a 9kg mass at x = 0, y = 1m. For this arrangement of the three masses, what is the angular acceleration of the system if a torque τ rotates the system about the y axis? All answers in inverse seconds squared.

Question

BlackTom AI · Accepted Answer

The problem asks for the angular acceleration about the y-axis for three masses placed in the xy-plane, given a torque τ about the y-axis. To analyze this, rewrite the situation in terms of rotational dynamics about the y-axis.

First, restating the setup: mass 1 is 1.2 kg at the origin (x=0, y=0). Mass 2 is 2.5 kg at (x=2 m, y=0). Mass 3 is 9 kg at (x=0, y=1 m). The axis of rotation is the y-axis, which runs vertically through the origin. The moment of inertia about the y-axis for a set of point masses is I_y = Σ m_i x_i^2 (since distance to the y-axis is the x-coordinate when z=0).

Option by option reasoning:

Option A: τ/9
This would imply I_y = 9. However, computing the actual I_y from the given positions, only the mass at x=2 m contributes, giving I_y = 2.5 × (2)^2 = 2.5 × 4 = 10 kg·m^2. Since this option would require I_y to be 9, it does not match the geometry of the setup. Therefore this option misidentifies the inertia about the y-axis.

Option B: zero
A nonzero torque about the y-axis will produce an angular acceleration unless the moment of inertia is infinite or the torque is somehow canceled by symmetry. Here, I_y is finite (10 kg·m^2) and the given torque τ is applied about the y-axis, so the angular acceleration is α_y = τ / I_y, not zero. Hence this choice is incorrect.

Option C: τ/10
From the coordinates, the only contribution to I_y comes from the mass at x=2 m, giving I_y = 2.5 × (2)^2 = 10 kg·m^2. Angular acceleration is α_y = τ / I_y = τ / 10. This matches the computed inertia and the standard relation α = τ / I about the rotation axis. This option aligns with the physical calculation.

Option D: τ/19
This would correspond to an inertia of I_y = 19, which does not arise from the given mass distribution (the only nonzero contribution to I_y is 10). This option incorrectly aggregates or weights the masses, leading to an incorrect inertia value.

Option E: τ/5
This would imply I_y = 5, which again does not match the actual inertia derived from the positions (only one mass contributes a nonzero term, giving 10). Thus this option underestimates the inertia and yields too large an angular acceleration.

In summary, the correct angular acceleration formula for this arrangement is α_y = τ / I_y with I_y = 10 kg·m^2, so α_y = τ / 10. The other options either miscompute the moment of inertia about the y-axis or misinterpret the effect of the torque on the rotational motion about that axis.

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