A \( 10 \, \text{kg} \) block on a horizontal surface (\( \mu_k = 0.35 \)) is pulled by a \( 7 \,
\text{kg} \) mass over a pulley. A \( 40 \, \text{N} \) force at \( 37^\circ \) to the horizontal
opposes the \( 10 \, \text{kg} \) block. What is the tension? (Take \( g = 10 \, \text{m/s}^2 \), \(
\sin 37^\circ = 0.6 \), \( \cos 37^\circ = 0.8 \))
For \( 7 \, \text{kg} \): \( 7g - T = 7a \Rightarrow 70 - T = 7a \).
For \( 10 \, \text{kg} \): \( T - f_k - F \cos 37^\circ = 10a \).
Normal force: \( N = mg + F \sin 37^\circ = 10 \times 10 + 40 \times 0.6 = 100 + 24 = 124 \, \text{N} \).
Friction: \( f_k = 0.35 \times 124 = 43.4 \, \text{N} \).
\( F \cos 37^\circ = 40 \times 0.8 = 32 \, \text{N} \).
Net force: \( T - 43.4 - 32 = 10a \Rightarrow T - 75.4 = 10a \).
Solve: \( 70 - T = 7a \), \( T - 75.4 = 10a \).
Substitute: \( 70 - (10a + 75.4) = 7a \Rightarrow 70 - 75.4 - 10a = 7a \Rightarrow -5.4 = 17a \).
\( a = \frac{-5.4}{17} \approx -0.318 \, \text{m/s}^2 \) (opposite direction, \( 7 \, \text{kg} \)
ascends).
\( T - 75.4 = 10 \times (-0.318) \Rightarrow T - 75.4 = -3.18 \Rightarrow T \approx 72.22 \, \text{N} \).