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A system of two wheels fixed to each other is free to rotate about a frictionless axis through the common center of the wheels and perpendicular to the page. Four forces are exerted tangentially to the rims of the wheels, as shown below.What is the magnitude of the net torque on the system about the axis? 1. τ = 2 F R 2. τ = 14 F R 3. τ = 0 4. τ = 5 F R 5. τ = F R

The image shows a system of two wheels (concentric), with forces applied tangentially on both wheels. There are two key radii marked: \(2R\) and \(3R\), and different forces acting at these radii. Let’s analyze the net torque acting on the system.

### Forces and Torques:
1. The **green circle** (inner wheel) has two forces acting on it:
   - **Force \(F\)** is applied tangentially at radius \(2R\), generating a torque of:
     \[
     \tau_1 = F \cdot 2R = 2FR
     \]
     (This is a counterclockwise torque.)
   
   - **Force \(F\)** is applied tangentially at the opposite side, but this force acts at radius \(2R\) as well, generating a torque:
     \[
     \tau_2 = F \cdot 2R = 2FR
     \]
     (This is a clockwise torque, opposing \(\tau_1\).)

2. The **blue circle** (outer wheel) has two forces acting on it:
   - **Force \(2F\)** is applied tangentially at radius \(3R\), generating a torque of:
     \[
     \tau_3 = 2F \cdot 3R = 6FR
     \]
     (This is a clockwise torque.)
   
   - **Force \(F\)** is applied tangentially at radius \(3R\), generating a torque of:
     \[
     \tau_4 = F \cdot 3R = 3FR
     \]
     (This is a clockwise torque.)

### Net Torque:
- Counterclockwise torques: \(2FR\)
- Clockwise torques: \(2FR + 6FR + 3FR = 11FR\)

So, the net torque is:
\[
\tau_{\text{net}} = 11FR - 2FR = 9FR
\]

However, the closest answer option is not provided directly, but based on this analysis the correct magnitude of the net torque would be \(9 F R\), even though that option isn't listed.

Do you need further clarification or details? Here are some additional questions to expand on this analysis:

1. How does the direction of torque affect rotational motion?
2. What would happen if the forces applied were different in magnitude?
3. How would the torque change if the radii were increased or decreased?
4. Can you relate this problem to angular acceleration and rotational dynamics?
5. What is the significance of forces being tangential in this setup?

**Tip:** Torque is always the product of force and the perpendicular distance from the axis of rotation, so always check both the force magnitude and its distance from the center.

A system of two wheels fixed to each other is free to rotate about a frictionless axis through the common center of the wheels and perpendicular to the page. Four forces are exerted tangentially to the rims of the wheels, as shown below. What is the magnitude of the net torque on the system about the axis? 1. τ = 2 F R 2. τ = 14 F R 3. τ = 0 4. τ = 5 F R 5. τ = F R

Explore the solution to a problem involving two connected wheels with tangential forces applied. Calculate the net torque using the formula τ = F * R, analyzing clockwise and counterclockwise torques for a system free to rotate about its axis.

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