Let's break down each part of the problem step-by-step: --- ## **a) Free-body diagram of the front wheel** **Forces acting on the front wheel:** - **Contact force from the ground at B:** This can be decomposed into a normal force \( N_B \) (upward) and possibly a horizontal friction force (not shown in the problem, so assumed zero). - **Force from the bicycle frame:** The frame exerts a downward force (from the mass at \( C \)), and the force \( F \) acts vertically downward at the pedal. - **Action-reaction pair:** The wheel exerts an upward force on the frame. **Diagram:** ``` | (Bicycle frame applies downward force) ↓ F + part of Mg ----- | | (Front wheel) ----- ↑ N_B (Ground reaction at B) ``` --- ## **b) Free-body diagram of the back wheel** **Forces acting on the back wheel:** - **Contact force from the ground at A:** Normal force \( N_A \) (upward). Typically, we neglect friction if not specified. - **Force from the bicycle frame:** The frame exerts a downward force (due to mass at \( C \)), but most of the weight is supported at the front. **Diagram:** ``` | (Bicycle frame applies downward force) ↓ part of Mg ----- | | (Back wheel) ----- ↑ N_A (Ground reaction at A) ``` --- ## **c) Free-body diagram of the entire bicycle** **Forces acting on the entire bicycle:** - **Weight \( Mg \) at C:** Acts vertically downward. - **Vertical downward force \( F \):** Applied at the pedal. - **Normal forces from the ground:** \( N_A \) at A (back wheel), \( N_B \) at B (front wheel). **Diagram:** ``` (C) (F at pedal) ↓ ↓ *---------* | | / \ (A) (B) ↑ N_A ↑ N_B ``` --- ### **Force directions:** - \( Mg \) at point \( C \), downward. - \( F \) at pedal, downward. - \( N_A \) at A, upward. - \( N_B \) at B, upward. --- ## **d) Assumptions in modeling the interaction force of the ground** - **Normal forces only:** The ground exerts only vertical normal forces at points A and B (\( N_A \) and \( N_B \)). - **No friction:** Unless otherwise specified, frictional forces are neglected. - **Point contacts:** The wheels are considered as point contacts with the ground (no area). - **Rigid bodies:** The wheels and frame are assumed to be rigid. - **Statics:** The system is in equilibrium (no acceleration). --- ## **Summary Table of Forces** | Force | Point of Application | Direction | |------------|---------------------|-------------| | \( Mg \) | C | Downward | | \( F \) | Pedal (front wheel) | Downward | | \( N_A \) | A (back wheel) | Upward | | \( N_B \) | B (front wheel) | Upward | --- ### **Final Answer Summary** 1. **Front wheel:** Upward normal from ground (\( N_B \)), downward force from frame (weight + pedal force component). 2. **Back wheel:** Upward normal from ground (\( N_A \)), downward force from frame (weight component). 3. **Entire bicycle:** Downward \( Mg \) at C, downward \( F \) at pedal, upward \( N_A \) at A, upward \( N_B \) at B. 4. **Assumptions:** Only vertical (normal) ground forces, no friction, point contact, rigid bodies, system in static equilibrium. Let me know if you'd like diagrams for each part!

# Bicycle with Unequal Wheels - Step-by-Step Solution ## **a) Free-body Diagram of the Front Wheel** ### **Forces Acting on the Front Wheel:** 1. **Contact Force from the Ground (B):** - Normal Force \( N_B \) (upward). - Potential horizontal friction force (assumed zero for simplicity). 2. **Force from the Bicycle Frame:** - Downward force due to the weight at point \( C \) and the force \( F \) applied at the pedal. ### **Diagram:** ``` | (Downward force from frame) ↓ F + part of Mg ----- | | (Front wheel) ----- ↑ N_B (Ground reaction at B) ``` --- ## **b) Free-body Diagram of the Back Wheel** ### **Forces Acting on the Back Wheel:** 1. **Contact Force from the Ground (A):** - Normal Force \( N_A \) (upward). 2. **Force from the Bicycle Frame:** - Downward force due to the weight at point \( C \) (less than at the front due to the bicycle’s tilt). ### **Diagram:** ``` | (Downward force from frame) ↓ part of Mg ----- | | (Back wheel) ----- ↑ N_A (Ground reaction at A) ``` --- ## **c) Free-body Diagram of the Entire Bicycle** ### **Forces Acting on the Entire Bicycle:** 1. **Weight \( Mg \):** - Acts vertically downward at point \( C \). 2. **Vertical Downward Force \( F \):** - Applied at the pedal. 3. **Normal Forces from the Ground:** - \( N_A \) at A (back wheel). - \( N_B \) at B (front wheel). ### **Diagram:** ``` (C) (F at pedal) ↓ ↓ *---------* | | / \ (A) (B) ↑ N_A ↑ N_B ``` --- ### **Force Directions:** - \( Mg \): Downward at C. - \( F \): Downward at the pedal. - \( N_A \): Upward at A. - \( N_B \): Upward at B. --- ## **d) Assumptions in Modeling Interaction Force of the Ground** 1. **Normal Forces Only:** The ground exerts only vertical normal forces (\( N_A \) and \( N_B \)). 2. **No Friction:** Frictional forces are neglected unless specified. 3. **Point Contacts:** Wheels are considered point contacts with the ground. 4. **Rigid Bodies:** Assumes wheels and frame are rigid. 5. **Statics:** The system is in static equilibrium (no acceleration). --- ## **Summary Table of Forces** | Force | Point of Application | Direction | |------------|---------------------|-------------| | \( Mg \) | C | Downward | | \( F \) | Pedal (front wheel) | Downward | | \( N_A \) | A (back wheel) | Upward | | \( N_B \) | B (front wheel) | Upward | --- ### **Final Answer Summary** 1. **Front Wheel:** - Upward normal force \( N_B \). - Downward forces \( F \) and part of \( Mg \). 2. **Back Wheel:** - Upward normal force \( N_A \). - Downward force from the frame (weight component). 3. **Entire Bicycle:** - Downward forces \( Mg \) at C and \( F \) at the pedal. - Upward forces \( N_A \) and \( N_B \). 4. **Assumptions:** - Only vertical normal forces, no friction, point contact, rigid bodies, system in static equilibrium. Feel free to ask for any further clarifications or additional diagrams!