A bicycle wheel has a radius R = 32 cm and a mass M = 1:80 kg which you may assume to be

concentrated on the outside radius. Assume that you apply a force F to a sprocket so that the wheel
has an angular acceleration ® = 4:50 rad/s2. A resistive force f = 131 N is applied to the rim of the
tire when it is moving.
calculate the strength of the required
force F, when the radius r is:
(a) 4.53 cm.

I worked out the problems using the sum of the Torque=I@

I@=-rFsin90+Rfsin90
Then I solved for F but my answer is 40 Newtons away from the right answer which is 940 N.
There is a picture of the problem in google if you just paste the entire problem on the search space it brings out a pdf file that has the picture it is problem 10-8

mr^2*4.5=r*F-131*.16

1.80*..32^2*4.5=.0453*F-131*.32

I get 940

To calculate the strength of the required force F when the radius r is 4.53 cm, you can follow these steps:

1. Convert the given radius R from centimeters to meters:
R = 32 cm = 0.32 m

2. Calculate the moment of inertia of the wheel using the mass M and the outer radius R:
I = M * R^2
= 1.80 kg * (0.32 m)^2
= 0.18432 kg·m^2

3. Convert the radius r from centimeters to meters:
r = 4.53 cm = 0.0453 m

4. Calculate the resistive torque exerted on the wheel:
τ_resistive = f * r
= 131 N * 0.0453 m
= 5.9343 N·m

5. Calculate the net torque acting on the wheel using the moment of inertia and the angular acceleration:
τ_net = I * α
= 0.18432 kg·m^2 * 4.50 rad/s^2
= 0.82944 N·m

6. Determine the torque applied by the force F by subtracting the resistive torque from the net torque:
τ_applied = τ_net - τ_resistive
= 0.82944 N·m - 5.9343 N·m
= -5.10486 N·m

7. Since the resistive force is applied in the opposite direction, the torque is negative. The force F also produces a torque in the opposite direction of the resistive torque. Therefore, the torque applied by F is:
τ_applied = -r * F
F = -τ_applied / r
= -(-5.10486 N·m) / 0.0453 m
= 112.7223 N

Therefore, the strength of the required force F when the radius r is 4.53 cm is approximately 112.7223 N.

To calculate the strength of the required force F when the radius r is 4.53 cm, you can use the following steps:

1. Convert the given values of radius and mass to SI units:
Radius R = 32 cm = 0.32 m
Mass M = 1.80 kg

2. Find the moment of inertia I of the bicycle wheel about its center using the formula:
I = 0.5 * M * R^2
Substituting the given values, we get:
I = 0.5 * 1.80 kg * (0.32 m)^2

3. Calculate the torque acting on the wheel due to the resistive force f:
Torque = r * f * sin(90°)
Substituting the given values and radius r = 4.53 cm = 0.0453 m, we obtain:
Torque = 0.0453 m * 131 N * sin(90°)

4. Use the net torque and the given angular acceleration to find the required force F:
Net Torque = I * angular acceleration (®)
Rearranging the formula, we have:
F = (I * ®) / r

5. Substituting the values of I, ®, and r, we can now calculate the required force F:
F = (0.5 * 1.80 kg * (0.32 m)^2 * 4.50 rad/s^2) / 0.0453 m

By following these steps, you should be able to calculate the correct value for the strength of the required force F.