TORQUE GRCC Physicspage 1

Angular Force (also known as “Torque”)

Physics 202

The kinetic energy of a rotating object is related to the angular velocity by the same relation that relates translational kinetic energy to linear velocity, providing mass is replaced by “rotational inertia” (which is sometimes also just called “moment of inertia”):

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IIImagine a system composed of a weight of mass M attached to one end of a lightweight (you can consider it massless) stick of length L that is free to rotate about the other end, as shown in the diagram below. The mass is initially at rest, and you should assume there are no forces (such as friction or gravity) in the direction of rotation until an external force is applied.The system may rotate about the pivot point, but the pivot will not move.

IIWhat is the rotational inertia (or moment of inertia) of this system?

IIIThe mass M may have a rather complicated linear acceleration due to forces from the stick and from the external force. Consider the component of the linear acceleration in the direction that contributes to angular acceleration about the pivot point. What does Newton’s second law tell us about the external force and this component of linear acceleration?

IIIIRewrite your answer from part 2 in terms of angular acceleration .

IVIWe may suspect that rotational dynamics will continue to mirror translational dynamics. If we want to write our equation for angular force and angular acceleration in terms of rotational inertia (so that it looks sort of like “F = m a”), we have to modify the equation from part 3 to get rotational inertia in there. What do you have to do to both sides of the equation? Write an equation based on your answer to part 3 that looks like:

“Angular Force” = (rotational inertia)  (angular acceleration)”

VIBased on the above, what is your first guess as to the form of angular force?

IIIINow try this one. The system we are trying to rotate is the same, but now we are pushing at a different angle, as in the diagram below:

IIIAgain, the mass M may have a complicated linear acceleration due to forces from the stick and from the external force. Consider the component of the linear acceleration in the direction that contributes to angular acceleration about the pivot point. Now what does Newton’s second law tell us about the external force and this component of linear acceleration?

IIIIRewrite your answer from above in terms of angular acceleration .

IIIIIWe still suspect that rotational dynamics will continue to mirror translational dynamics. Again, write an equation based on your answer to part 3 that looks like:

“Angular Force” = (rotational inertia)  (angular acceleration)”

IVIIIf all has gone well you have justrewritten Newton’s 2ndlaw in rotational form (in the case where rotational inertia is constant).Based on the above, what is your refined guess as to the form of angular force?

IIIIII“Angular force” is usually called“torque” and given the symbol .

IIIIBased on parts Iand II, what are the units for torque?

(Note: we never report torque in units of Joules, even if it looks like Joules.)

IIIIIWe can also check the units by thinking about “rotational work” (and this also allows us to check our rotational form of Newton’s 2ndlaw). Conservation of energy tells us that we expect“” or more generally,

Do your units for torque agree with this equation?

IVIVThe equation for work in the case of constant torque provides a quick (and easy?) check of our equation for rotational force and acceleration. If torque is constant and there is no change in potential energy, then we have

IIVIf both sides of the above equation are divided by t we get an equation for angular power. Write out this equation, and rewrite it in the limit that .

IIIVThe left-hand side of your equationabove should be fairly simple. The right-hand side should involve taking a derivative with respect to time. Assuming the object we are rotating is rigid, what can be immediately pulled out of the derivative?

IIIIVAt this point you should be left with an expression involving. Expand this expression using the chain rule. You should get something involving .

Now write out your complete equation from part 1, expanded and simplified as much as possible. Can you cancel any terms on both sides of the equal sign? What’s left?