Three Rotating Objects with Identical Torque (1)

The work done by the pull of the string is force times distance. Assuming no air resistance or frictional loses, all of this work goes into increasing the rotational kinetic energy of the objects.

If the strings were pulled for the same distance , the three objects would end up with the same kinetic energy and the last answer would be correct. But the question states that the string was pulled for the same time in each case.

The three objects spin up with different speeds. The object which spins up fasted unwinds more string, so that it has the greatest Force x Distance and so the greatest energy.

The resistance to a torque's effort to produce rotational acceleration is given by the moment of inertia of the object. The moments of inertia for the disk, sphere and hoop are respectively $\frac{mr^2}{2}$ , $\frac{2mr^2}{5}$ and $mr^2$

So the sphere has the lowest moment of inertia, so it speeds up fastest, so it unwinds the greatest length of string, so the sphere's force does the most work, and so the sphere ends up with the greatest kinetic energy!

The torque for all three objects is $F*R$ . By the rotational form of Newtons second law we get that the angular acceleration $α=$ $\frac{F*R}{I}$ where I is the moment of inertia. After a time interval t the tangential velocity v will be $v=$ $\frac{F*R*t}{I}$ . If we divide this by R we get the angular velocity $w=$ $\frac{F*t}{I}$ . The rotational kinetic energy is given by $K=$ $\frac{1}{2}$ $Iw^2$ . If we insert w into this we get $K=$ $\frac{F^2*t^2}{2I}$ . The force and time is equal in all scenarioes. Therefore the object with the greatest kinetic energy will have the smallest moment of inertia. Of the three objects the sphere has the smallest moment ofinertia. Therefore is the sphere the correct answer.

Each of the three objects is subject to the same net torque because the force in each case is the same, and the radius of each object is the same. Since the time intervals are the same, and the angular impulse is defined by torque times time, the change in each object's angular momentum will be the same. Therefore, the object with the smallest rotational inertia will have the largest angular velocity at the end of the interval. The solid sphere has the smallest rotational inertia of the three objects; its rotational inertia is $\frac{2}{5}mR^{2}$ . The formula for rotational kinetic energy depends on the square of the angular velocity ( $K = \frac{1}{2}I\omega^{2}$ ) so the sphere will have the most rotational kinetic energy.

Being ignorant of the formulas for rotational inertia, here's how I did it. We remember $E = m c^2$ , where c is velocity (of light in this case, but we also know the relationship applies to other velocities). So energy is proportional to mass, but proportional to the square of velocity. Assuming constant density within each shape (which probably should have been made explicit in the problem), the sphere (compared to hoop and disc) has less of its mass concentrated near the edge, where the force is applied and where the leverage is greatest. So the velocity of bits on its edge (considered linear at the point of force application) will increase more than that of the other shapes. The lower mass decreases the resulting energy by a certain factor, but the higher velocity increases the resulting energy by the same factor squared.

The physical hint is the more sphere the easier to spin( thanks to momen inertia(I) ), so it is easy to assume that the sphere must be the fastest at the last in a same time interval, therefore, the greatest rotational kinetic energy must be sphere

Greater Volume gives the most momentum, I tried to intuit. Thought the wheel shape moves most easily, the disc secondly and the sphere had the most inertia to overcome. But Mass same and frictionless so while I first thought the wheel would spin the most so it would have the most kinetic energy, but then thought it would be dissipated in easy exchange....so the sphere might capture the most energy and transfer through to it's kinetics- with all other things being equalized through applied force. My daughter then told me that generally k=(\frac{mv^2}{2}), which seemed to suggest the greater volume would have the greater (Ek)

After reading some stuff on Wikipedia, i found that E{rot}= .5 * Angular Velocity * Angular Mass, where Angular Velocity stays the same at all 3 objects, since, when you pull at 3 objects which have the same mass with the same force, they will all rotate at the same velocity, and where the Angular Mass is idk but read from wikipedia that it gets smaller as the mass is closer to the center of the object. And now, if you take a wheel or a disc, the ratio of mass is less extreme then if you would take a sphere, where there is much more mass much further away from the center.

the total kinetic energy of rotating body is given by 1/2 I w^2 so for moment of inertia is the only variable quantity in above eqn and because solid sphere has higher moment of inertia, it has high KE

Given same angular impulses (that is force × radius × time), the fastest (and therefore with greatest energy since they all have same mass) object will be the one with fewest momentum of inertia. You don't have to know exact formulas except the definition: $I=\displaystyle \sum m_ir_i^2$ where $r$ is distance to the axis.

Knowing that, you can determine which object has fewer $I$ : hoop has all its mass at (nearly) $R$ , and sphere has more mass closer to the axis than the disc because it is volumetric (if difficult to imagine, use the word "points" instead of "mass", but technically it is quite inscientific). Hence, the sphere has the lowest momentum of inertia, and would have greatest energy in this experiment.

First time commenting here. Don't know if it's a valid argument, but since all objects have the same mass, and because the sphere has a greater area, the sphere would be lighter in proportion, making it easier to rotate.