NEET Laws of Motion — Set 8

Question 1

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 $)

Accepted Answer

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 $.

Question 2

A $ 5 \, \text{kg} $ block on a $ 37^\circ $ incline ($ \mu_s = 0.4 $) is pulled upward by a force at $ 30^\circ $ to the horizontal just sufficient to start motion. What is the force? (Take $ g = 10 \, \text{m/s}^2 $, $ \sin 37^\circ = 0.6 $, $ \cos 37^\circ = 0.8 $, $ \sin 30^\circ = 0.5 $, $ \cos 30^\circ = 0.866 $)

Accepted Answer

The block is on the verge of moving upward, so the net force along the incline is zero, and friction acts downward. Along incline: $ F \cos (37^\circ - 30^\circ) - mg \sin 37^\circ - f_s = 0 $, where $ \theta = 7^\circ $. Approximate: $ \cos 7^\circ \approx 0.992 $, $ \sin 7^\circ \approx 0.122 $ (for simplicity, use exact if needed). Normal force: $ N = mg \cos 37^\circ - F \sin (37^\circ - 30^\circ) $. $ mg \sin 37^\circ = 5 \times 10 \times 0.6 = 30 \, \text{N} $, $ mg \cos 37^\circ = 5 \times 10 \times 0.8 = 40 \, \text{N} $. $ N = 40 - F \times 0.122 $, $ f_s = 0.4 (40 - 0.122F) $. Substitute: $ F \times 0.992 - 30 - 0.4 (40 - 0.122F) = 0 $. $ 0.992F - 30 - 16 + 0.0488F = 0 \Rightarrow 1.0408F - 46 = 0 $. $ 1.0408F = 46 \Rightarrow F \approx 44.2 \, \text{N} $.

Question 3

A $ 8 \, \text{kg} $ block on a horizontal surface ($ \mu_k = 0.3 $) is pulled by a $ 5 \, \text{kg} $ mass over a pulley. What is the tension in the string? (Take $ g = 10 \, \text{m/s}^2 $)

Accepted Answer

For $ 5 \, \text{kg} $: $ 5g - T = 5a \Rightarrow 50 - T = 5a $. For $ 8 \, \text{kg} $: $ T - f_k = 8a $. Normal: $ N = mg = 8 \times 10 = 80 \, \text{N} $. Friction: $ f_k = 0.3 \times 80 = 24 \, \text{N} $. Net force: $ T - 24 = 8a $. Solve: $ 50 - T = 5a $, $ T - 24 = 8a $. Substitute: $ 50 - (8a + 24) = 5a \Rightarrow 50 - 24 - 8a = 5a \Rightarrow 26 = 13a $. $ a = 2 \, \text{m/s}^2 $. $ T - 24 = 8 \times 2 \Rightarrow T - 24 = 16 \Rightarrow T = 40 \, \text{N} $.

NEET Physics: Laws of Motion — Practice Set 8

Q3. A \( 8 \, \text{kg} \) block on a horizontal surface (\( \mu_k = 0.3 \)) is pulled by a \( 5 \, \text{kg} \) mass over a pulley. What is the tension in the string? (Take \( g = 10 \, \text{m/s}^2 \))

Q4. A \( 0.18 \, \text{kg} \) ball moving at \( 20 \, \text{m/s} \) at \( 37^\circ \) to a wall rebounds at the same speed and angle. What is the impulse perpendicular to the wall? (Take \( \sin 37^\circ = 0.6 \), \( \cos 37^\circ = 0.8 \))

Q5. A \( 7 \, \text{kg} \) block on a \( 60^\circ \) incline (\( \mu_k = 0.2 \)) is connected to a \( 4 \, \text{kg} \) mass over a pulley. What is the tension? (Take \( g = 10 \, \text{m/s}^2 \), \( \sin 60^\circ = 0.866 \), \( \cos 60^\circ = 0.5 \))

Q6. A \( 2 \, \text{kg} \) block on a \( 37^\circ \) incline (\( \mu_k = 0.3 \)) accelerates down at \( 2 \, \text{m/s}^2 \). What external force acts parallel to the incline? (Take \( g = 10 \, \text{m/s}^2 \), \( \sin 37^\circ = 0.6 \), \( \cos 37^\circ = 0.8 \))

Q7. A \( 4 \, \text{kg} \) mass on a surface (\( \mu_k = 0.2 \)) is pulled by a \( 6 \, \text{kg} \) mass over a pulley. If a \( 5 \, \text{N} \) force opposes the \( 4 \, \text{kg} \) mass, what is the tension? (Take \( g = 10 \, \text{m/s}^2 \))

Q9. A constant retarding force of \( 20 \, \text{N} \) stops a body of mass \( 4 \, \text{kg} \) moving at \( 10 \, \text{m/s} \). How long does it take to stop?

Q10. A \( 0.4 \, \text{kg} \) ball at \( 50 \, \text{m/s} \) hits a wall at \( 45^\circ \) and rebounds at \( 60^\circ \) with speed \( 40 \, \text{m/s} \). What is the impulse along the wall? (Take \( \sin 45^\circ = \cos 45^\circ = 0.707 \), \( \sin 60^\circ = 0.866 \), \( \cos 60^\circ = 0.5 \))