NEET Laws of Motion — Set 9

Question 1

A $ 4 \, \text{kg} $ block on a horizontal surface ($ \mu_k = 0.3 $) is connected to a $ 5 \, \text{kg} $ mass over a pulley. A force of $ 30 \, \text{N} $ at $ 53^\circ $ to the horizontal pulls the $ 4 \, \text{kg} $ block, and a $ 10 \, \text{N} $ force opposes the $ 5 \, \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

The system accelerates as the $ 5 \, \text{kg} $ mass moves downward and the $ 4 \, \text{kg} $ block moves horizontally. For the $ 5 \, \text{kg} $ mass (vertical): Forces are weight downward, tension upward, and opposing force upward. Net force: $ 5g - T - 10 = 5a \Rightarrow 50 - T - 10 = 5a \Rightarrow 40 - T = 5a $. For the $ 4 \, \text{kg} $ block (horizontal): Forces are tension, kinetic friction, and the horizontal component of the $ 30 \, \text{N} $ force. Normal force: $ N = mg - F \sin 53^\circ = 4 \times 10 - 30 \times 0.8 = 40 - 24 = 16 \, \text{N} $. Friction: $ f_k = \mu_k N = 0.3 \times 16 = 4.8 \, \text{N} $. Horizontal component of force: $ F \cos 53^\circ = 30 \times 0.6 = 18 \, \text{N} $. Net force: $ T + 18 - 4.8 = 4a \Rightarrow T + 13.2 = 4a $. Solve the two equations: $ 40 - T = 5a $ and $ T + 13.2 = 4a $. Substitute $ T = 40 - 5a $ into the second: $ 40 - 5a + 13.2 = 4a $. Combine: $ 53.2 = 4a + 5a \Rightarrow 53.2 = 9a $. Solve: $ a = \frac{53.2}{9} \approx 5.91 \, \text{m/s}^2 $.

Question 2

A $ 0.4 \, \text{kg} $ stone is whirled in a horizontal circle of radius $ 1.8 \, \text{m} $ with a tension of $ 15 \, \text{N} $. What is the speed? (Take $ g = 10 \, \text{m/s}^2 $)

Accepted Answer

Tension equals centripetal force: $ T = m v^2 / r $. Substitute: $ 15 = 0.4 \times v^2 / 1.8 $. $ 15 = 0.222 v^2 \Rightarrow v^2 = \frac{15}{0.222} \approx 67.57 $. $ v = \sqrt{67.57} \approx 8.22 \, \text{m/s} $.

Question 3

A $ 5.5 \, \text{kg} $ block on a $ 37^\circ $ incline ($ \mu_k = 0.2 $) is pulled upward by a $ 7.5 \, \text{kg} $ mass over a pulley. What is the acceleration? (Take $ g = 10 \, \text{m/s}^2 $, $ \sin 37^\circ = 0.6 $, $ \cos 37^\circ = 0.8 $)

Accepted Answer

For $ 7.5 \, \text{kg} $: $ 7.5g - T = 7.5a \Rightarrow 75 - T = 7.5a $. For $ 5.5 \, \text{kg} $: $ T - mg \sin 37^\circ - f_k = 5.5a $. $ N = mg \cos 37^\circ = 5.5 \times 10 \times 0.8 = 44 \, \text{N} $. Friction: $ f_k = 0.2 \times 44 = 8.8 \, \text{N} $. $ mg \sin 37^\circ = 55 \times 0.6 = 33 \, \text{N} $. Net force: $ T - 33 - 8.8 = 5.5a \Rightarrow T - 41.8 = 5.5a $. Solve: $ 75 - T = 7.5a $, $ T - 41.8 = 5.5a $. Substitute: $ 75 - (5.5a + 41.8) = 7.5a \Rightarrow 75 - 41.8 - 5.5a = 7.5a \Rightarrow 33.2 = 13a $. $ a = \frac{33.2}{13} \approx 2.55 \, \text{m/s}^2 $.

NEET Physics: Laws of Motion — Practice Set 9

Q2. A \( 0.4 \, \text{kg} \) stone is whirled in a horizontal circle of radius \( 1.8 \, \text{m} \) with a tension of \( 15 \, \text{N} \). What is the speed? (Take \( g = 10 \, \text{m/s}^2 \))

Q3. A \( 5.5 \, \text{kg} \) block on a \( 37^\circ \) incline (\( \mu_k = 0.2 \)) is pulled upward by a \( 7.5 \, \text{kg} \) mass over a pulley. What is the acceleration? (Take \( g = 10 \, \text{m/s}^2 \), \( \sin 37^\circ = 0.6 \), \( \cos 37^\circ = 0.8 \))

Q4. A block of mass \( 5 \, \text{kg} \) is suspended by a rope from the ceiling. What is the tension in the rope? (Take \( g = 10 \, \text{m/s}^2 \))

Q6. A \( 0.42 \, \text{kg} \) stone in a vertical circle of radius \( 1.4 \, \text{m} \) has a speed of \( 12 \, \text{m/s} \) at the bottom. What is the tension at the top? (Take \( g = 10 \, \text{m/s}^2 \))

Q7. A \( 1050 \, \text{kg} \) car turns on a banked road (\( \theta = 20^\circ \)) with radius \( 60 \, \text{m} \) at the optimum speed to avoid friction. What is the speed? (Take \( g = 10 \, \text{m/s}^2 \), \( \tan 20^\circ \approx 0.364 \))

Q8. A \( 2200 \, \text{kg} \) rocket accelerates upward at \( 3.5 \, \text{m/s}^2 \) with a thrust of \( 32000 \, \text{N} \). What is the mass ejection rate if the gas speed is \( 80 \, \text{m/s} \) relative to the rocket? (Take \( g = 10 \, \text{m/s}^2 \))

Q9. A \( 2000 \, \text{kg} \) truck accelerates at \( 2.2 \, \text{m/s}^2 \) for \( 5 \, \text{s} \), then a \( 1.4 \, \text{kg} \) stone is dropped. What is the horizontal distance traveled by the stone in \( 2 \, \text{s} \)? (Neglect air resistance)

Q10. A \( 950 \, \text{kg} \) car turns on a banked road (\( \theta = 25^\circ \), \( \mu_s = 0.2 \)) with radius \( 40 \, \text{m} \). What is the maximum speed without slipping? (Take \( g = 10 \, \text{m/s}^2 \), \( \tan 25^\circ \approx 0.466 \))