Let's solve the problem step by step. --- ### **Step 1: Draw Free Body Diagrams and Write Force Equilibrium Equations** **For Block \( m_1 \) on the incline:** - **Gravity**: \( m_1 g \) (acts downward) - Component parallel to incline: \( m_1 g \sin\theta \) - Component perpendicular to incline: \( m_1 g \cos\theta \) - **Normal force**: \( N = m_1 g \cos\theta \) - **Friction force**: \( f_s \leq \mu_s N = \mu_s m_1 g \cos\theta \) - **Tension**: \( T \) (up the incline, due to hanging mass) **For Hanging Mass \( m_2 \):** - Gravity: \( m_2 g \) (down) - Tension: \( T \) (up) **Equilibrium equations:** - For \( m_1 \) (along the incline): \( T = m_1 g \sin\theta \pm f_s \) - For \( m_2 \): \( T = m_2 g \) --- #### **Step 1A: Maximum \( m_2 \) (block about to move up incline)** Friction acts **down** the incline (opposing upward motion): \[ T = m_1 g \sin\theta + f_{s,\text{max}} \] \[ m_2 g = m_1 g \sin\theta + \mu_s m_1 g \cos\theta \] \[ m_2 = m_1 (\sin\theta + \mu_s \cos\theta) \] **Explanation:** This is the maximum value of \( m_2 \) before the block slides **up** the incline. --- #### **Step 1B: Minimum \( m_2 \) (block about to move down incline)** Friction acts **up** the incline (opposing downward motion): \[ T = m_1 g \sin\theta - f_{s,\text{max}} \] \[ m_2 g = m_1 g \sin\theta - \mu_s m_1 g \cos\theta \] \[ m_2 = m_1 (\sin\theta - \mu_s \cos\theta) \] **Explanation:** This is the minimum value of \( m_2 \) before the block slides **down** the incline. --- ### **Step 2: Plug in the Given Values** Given: - \( \theta = 26^\circ \) - \( \mu_s = .36 \) Calculate: - \( \sin 26^\circ \approx .4384 \) - \( \cos 26^\circ \approx .8988 \) **Minimum \( m_2 \):** \[ m_2 = m_1 [.4384 - .36 \times .8988] \] \[ m_2 = m_1 [.4384 - .3236] = m_1 (.1148) \] **Maximum \( m_2 \):** \[ m_2 = m_1 [.4384 + .36 \times .8988] \] \[ m_2 = m_1 [.4384 + .3236] = m_1 (.762) \] --- ### **Final Answer** \[ \boxed{(.1148\, m_1) \leq m_2 \leq (.762\, m_1)} \] **Explanation:** For the system to remain in equilibrium, the hanging mass \( m_2 \) must be within this range. If it's less, the block will slide down; if it's more, the block will slide up.