A \( 12 \, \text{kg} \) block on a horizontal surface (\( \mu_k = 0.3 \)) is pulled by a \( 10 \,
\text{kg} \) mass over a pulley. A \( 35 \, \text{N} \) force at \( 53^\circ \) opposes the \( 12 \,
\text{kg} \) block, and a \( 20 \, \text{N} \) force aids the \( 10 \, \text{kg} \) mass. What is the
acceleration? (Take \( g = 10 \, \text{m/s}^2 \), \( \sin 53^\circ = 0.8 \), \( \cos 53^\circ = 0.6 \))
For \( 10 \, \text{kg} \): \( 10g + 20 - T = 10a \Rightarrow 100 + 20 - T = 10a \Rightarrow 120 - T = 10a
\).
For \( 12 \, \text{kg} \): \( T - f_k - F \cos 53^\circ = 12a \).
Normal: \( N = mg + F \sin 53^\circ = 12 \times 10 + 35 \times 0.8 = 120 + 28 = 148 \, \text{N} \).
Friction: \( f_k = 0.3 \times 148 = 44.4 \, \text{N} \).
\( F \cos 53^\circ = 35 \times 0.6 = 21 \, \text{N} \).
Net force: \( T - 44.4 - 21 = 12a \Rightarrow T - 65.4 = 12a \).
Solve: \( 120 - T = 10a \), \( T - 65.4 = 12a \).
Substitute: \( 120 - (12a + 65.4) = 10a \Rightarrow 120 - 65.4 - 12a = 10a \Rightarrow 54.6 = 22a \).
\( a = \frac{54.6}{22} \approx 2.48 \, \text{m/s}^2 \).