# Solution: Rate of Slip at Point A Let's break down the problem and solve for the rate of slip at the contact point **A**. ## Given Data - Linear velocity of the wheel's center (\( v_C \)): **2 ft/s** (up the incline) - Angular velocity (\( \omega \)): **5 rad/s** (clockwise) - Radius (\( r \)): **15 in** = **1.25 ft** (since 1 ft = 12 in) ## Step 1: Velocity at Point A The velocity of point A (the point in contact with the surface) relative to the ground is: \[ \vec{v}_A = \vec{v}_C + \vec{v}_{A/C} \] Where: - \( \vec{v}_C \) = velocity of the center (2 ft/s, up the incline) - \( \vec{v}_{A/C} = \omega \times r \) (perpendicular to the radius, due to rotation) ### Direction of Rotational Velocity - The wheel spins **clockwise**, so at point A (bottom of the wheel), the rotational velocity due to \(\omega\) is **down the incline**. ### Magnitude of Rotational Velocity \[ v_{A/C} = \omega \times r = 5 \, \text{rad/s} \times 1.25 \, \text{ft} = 6.25 \, \text{ft/s} \] - This is **down the incline**. ## Step 2: Combine Velocities - \( v_C \) is **up the incline** (**positive** direction). - \( v_{A/C} \) is **down the incline** (**negative** direction). So, \[ v_A = v_C - v_{A/C} \] \[ v_A = 2 \, \text{ft/s} - 6.25 \, \text{ft/s} = -4.25 \, \text{ft/s} \] - **Negative sign** means the velocity at point A is **down the incline** relative to the surface. ## Step 3: Rate of Slip The **rate of slip** is the **magnitude** of the velocity of point A relative to the surface: \[ \boxed{4.25 \, \text{ft/s}} \] This is the relative velocity between point A on the wheel and the surface (down the incline). --- ### **Summary Table** | Quantity | Value | Direction | |---------------|---------------|------------------| | \(v_C\) | 2 ft/s | Up the incline | | \(\omega\) | 5 rad/s | Clockwise | | \(r\) | 1.25 ft | — | | \(v_{A/C}\) | 6.25 ft/s | Down the incline | | **Slip Rate** | **4.25 ft/s** | Down the incline | --- ## **Final Answer** > **The rate of slip at the contact point A is \( \boxed{4.25 \, \text{ft/s}} \) down the incline.**