Question

A flat, solid cylindrical grinding wheel with a diameter of 24.2 cm is spinning at 3000 rpm when its power is suddenly turned off. A workman continues to press his tool bit toward the wheel's center at the wheel's circumference so as to continue to grind as the wheel coasts to a stop.

If the wheel has a moment of inertia of 4.99 kg⋅m2 , the necessary torque (116 m*N ) must be exerted by the workman to bring it to rest in 13.5 s .

If the coefficient of kinetic friction between the tool bit and the wheel surface is 0.800, how hard must the workman push on the bit?(in newtons)

Answer 1

To understand this problem we must know that this is a torque problem. Then we need to know that the grinding wheel have his own rotational torque and when the tool is pressed against the wheel surface it creates a frictional force that combined with the wheel's radius gives us another torque opposite to the one from the wheel.

So, basically we need to make the wheel's rotational torque equal to the frictional force torque because, since both are opposite, will bring us to equilibrium (the wheel fully stopped).

Now the 2 torque equations that come into this are:

This is the torque created by the frictional force and the radius of the wheel.

This is the torque that the wheel have which relates to the moment of inertia and the angular acceleration. This torque is given in the problem (116 Nm)

So we must satisfy:

And since we already know that the right portion of the equation is 116Nm we have

So we need to find which is being F the force applied by the workman with his tool (What we looking for), So:

Then:

And thats our answer.

BONUS INFORMATION

If we didnt have the torque that the worker needed to excert (116Nm) we could have calculated it using the other given information, rpm and time. To do that we had to only calculate the angular acceleration

To do this we can use:

On that equation the (since the wheel will stop) and to calculate we convert the 3000rpm to rad/s to get the so:

So since

And this gives us:

And thats how we could have gotten that Torsion that was given to us to calculate the Force.