Explanation of net torque in a problem

[jolt527]

Explanation of "net torque" in a problem

I hate to ask such a simple question, but I was working on a problem where the idea of "net torque" was calculated differently than I thought. I more or less and just looking for the flaw in my logic and an explanation.

Here's the main problem: A bicycle wheel has a diameter of 64.0 cm and a mass of 1.80 kg. Assume that the wheel is a hoop with all of its mass concentrated on the outside radius. The bicycle is placed on a stationary stand on rollers, and a resistive force of 120 N is applied tangent to the rim of the tire. What force must be applied by a chain passing over a 9.00-cm-diameter sprocket if the wheel is to attain an acceleration of 4.50 rad/s^2?

Okay, most of what I worked out was correct except for one part of the final equation. Here's some work that produces the correct answer:

$$\Sigma \tau = I \alpha = M_{W}R_{W}^2 \alpha$$
$$M_{W}R_{W}^2 \alpha = \Sigma rF \sin \phi = R_{SPR}F_{CH} - R_{W}F_{RES}$$

Solving for the force of the chain $$\left(F_{CH}\right)$$:

$$F_{CH} = \frac{M_{W}R_{W}^2 \alpha + R_{W}F_{RES}}{R_{SPR}} = \textrm{872 N}$$

Now, the only problem is with part of the net torque, namely the part dealing with the sprocket $$\left( R_{SPR}F_{CH} \right)$$ on the second line of equations. When we are dealing with the net torque on something, I supposed that that term ought to be $$\left( R_{W}F_{CH} \right)$$ because we're talking about the force of the chain on the wheel. I'm stumped on why we're using the radius of the sprocket instead of the radius of the wheel. Is my understanding of what net torque means wrong, or am I just misunderstanding it for the problem? Any help is greatly appreciated, and thank you!

-Keith

 

[cepheid]

I supposed that that term ought to be $$\left( R_{W}F_{CH} \right)$$ because we're talking about the force of the chain on the wheel. I'm stumped on why we're using the radius of the sprocket instead of the radius of the wheel.

-Keith

Hi Keith,

Look at a bicycle. When you pedal, a sprocket attached to the pedals rotates. This sprocket has a chain looped around it. The other end of the chain is looped around a sprocket that is attached to the back wheel at its centre . This latter sprocket is the sprocket in question.

So, when you pedal a bike, you apply a torque to the sprocket (or gear) on the back wheel. This torque is in turn applied to the back wheel, but only by virtue of the fact that they are attached. The diameter of the sprocket is the only lever arm available for the force from the chain. The full diameter of the wheel is not available, because that's not what the chain is pulling on.

 

[jolt527]

Thanks for the reply, cepheid! I actually feel really dumb now, because I was picturing the sprocket away from the wheel instead of attached to it. Now the mathematics make total sense, along with the answer. Sorry to have wasted some of your time, but thanks for helping me to know that my physics are okay, just my understanding of the problem was off.

-Keith

 

[cepheid]

No problems Keith. Without a picture, and only a description, this problem can be a bit abstract!