Math Problem Statement

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

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

Solution

The image shows a system of two wheels (concentric), with forces applied tangentially on both wheels. There are two key radii marked: 2R2R and 3R3R, 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 FF is applied tangentially at radius 2R2R, generating a torque of: τ1=F2R=2FR\tau_1 = F \cdot 2R = 2FR (This is a counterclockwise torque.)

    • Force FF is applied tangentially at the opposite side, but this force acts at radius 2R2R as well, generating a torque: τ2=F2R=2FR\tau_2 = F \cdot 2R = 2FR (This is a clockwise torque, opposing τ1\tau_1.)

  2. The blue circle (outer wheel) has two forces acting on it:

    • Force 2F2F is applied tangentially at radius 3R3R, generating a torque of: τ3=2F3R=6FR\tau_3 = 2F \cdot 3R = 6FR (This is a clockwise torque.)

    • Force FF is applied tangentially at radius 3R3R, generating a torque of: τ4=F3R=3FR\tau_4 = F \cdot 3R = 3FR (This is a clockwise torque.)

Net Torque:

  • Counterclockwise torques: 2FR2FR
  • Clockwise torques: 2FR+6FR+3FR=11FR2FR + 6FR + 3FR = 11FR

So, the net torque is: τnet=11FR2FR=9FR\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 9FR9 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.

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Math Problem Analysis

Mathematical Concepts

Torque
Rotational Motion
Forces

Formulas

Torque (τ) = Force (F) * Radius (R)
Net torque = sum of all torques (τ_net = Στ_clockwise - Στ_counterclockwise)

Theorems

Newton's Second Law for Rotation

Suitable Grade Level

Grades 11-12, College