Two uniform bars of the same dimensions are constructed from the same material. One bar has five evenly spaced holes through it and the second bar has only two holes. In these cases, the bars are slid over vertical pegs and rest on a horizontal surface, where friction between the bar and the surface is negligible. The two bars are each pulled by horizontal forces of equal magnitude F from their right end as shown above. The bars’ resulting angular accelerations are recorded.

Question

Two uniform bars of the same dimensions are constructed from the same material. One bar has five evenly spaced holes through it and the second bar has only two holes. In these cases, the bars are slid over vertical pegs and rest on a horizontal surface, where friction between the bar and the surface is negligible. The two bars are each pulled by horizontal forces of equal magnitude F from their right end as shown above. The bars’ resulting angular accelerations are recorded.

Is the magnitude of the initial angular acceleration of the bar in Case 1 larger than, smaller than, or equal to the magnitude of the initial angular acceleration of the bar in Case 2? Explain your reasoning.

Answer 1

To determine whether the magnitude of the initial angular acceleration of the bar in case 1 is larger, smaller, or equal to the magnitude of the initial angular acceleration of the bar in case 2, we need to understand the factors that affect the angular acceleration of an object.

The angular acceleration of an object is directly proportional to the torque applied to the object and inversely proportional to its moment of inertia. Torque (τ) is the product of the force (F) applied perpendicular to the axis of rotation and the radius (r) at which the force is applied. Mathematically, τ = F * r.

In this case, the two bars have the same dimensions and are made of the same material. Therefore, their moments of inertia will be the same. The moment of inertia depends on the mass distribution and shape of the object, which is constant for both bars.

The only difference between the two bars is the number of holes. The presence of holes reduces the mass distribution of the bar, effectively reducing its moment of inertia. In case 1, the bar with five evenly spaced holes has a lower moment of inertia compared to the bar in case 2, which only has two holes.

Given that both bars experience equal magnitude forces (F) applied at the same radius (r), the bar with a lower moment of inertia (case 1) will experience a larger torque (τ) and therefore a larger angular acceleration compared to the bar in case 2.

Hence, the magnitude of the initial angular acceleration of the bar in case 1 is larger than the magnitude of the initial angular acceleration of the bar in case 2.