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.

Solution :

The angular acceleration, is obtained from the equation of the of rotational motion,

Thus,

or

where is torque, F is force, d is moment arm distance, I is the moment of inertia

Thus,

Now if the force and the moment arm distance are constant, then the

That is when, F = d = constant, then   .

Thus, moment of inertia, I is proportional to mass of the bar.

The mass is less for the bar in case (1) in comparison with that with the bar in case (2) due to the holes that is made in the bar.

Therefore, the bar in case (1), has less moment of inertia and a greater angular acceleration.

the motion of a pendulum is a classic example of mechanical energy conservation. a pendulum consists of a mass (known as a bob ) attached by a string to a pivot point. as the pendulum moves it sweeps out a circular arc, moving back and forth in a periodic fashion.

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

where is unknown. because the position is parabolic, there is symmetry about the midpoint at , and we know that

so that the position function is

which means the velocity is

then the total distance traveled is

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at and ends at , so its displacement is 0.