# Worksheet: Applications of Newtonβs Second Law: Inclined Pulley

In this worksheet, we will practice solving problems on the motion of two bodies connected by a string passing over a smooth pulley with one of them on an inclined plane.

**Q1: **

A body of mass 8.1 kg was resting on a smooth plane inclined at an angle of to the horizontal where . The body was connected to the end of a string passing through a pulley fixed at the top of the plane. At the other end of the string, a body weighing 26.9 kg was hanging freely. The system was released from rest and started moving, and 2 seconds later, the string snapped. Find the distance the body on the plane ascended after the string broke and before it momentarily came to rest. Take .

**Q2: **

A body of mass 5 kg rests on a smooth plane inclined at an angle of to the horizontal. It is connected by a light inextensible string passing over a smooth pulley fixed at the top of the plane, to another body of mass 19 kg hanging freely vertically below the pulley. Give that the acceleration due to gravity , determine the acceleration of the system.

**Q3: **

A body of mass 16 kg rests on a smooth plane inclined at to the horizontal. It is connected, by a light inextensible string passing over a smooth pulley fixed to the top of the plane, to another body of the same mass hanging freely vertically below the pulley. Given that the acceleration due to gravity , determine the tension in the string.

**Q4: **

Two bodies of equal masses of 7.4 kg are connected by a light inelastic string. One of the bodies rests on a smooth plane inclined at to the horizontal. The string passes over a smooth pulley fixed at the top of the plane, and the other body is left to hang freely vertically below the pulley. Find the force acting on the pulley when the system is released from rest. Take the acceleration due to gravity to be .

**Q5: **

A body of mass g rests on a smooth plane inclined at an angle of to the horizontal. It is connected, by a light string passing over a smooth pulley fixed at the top of the plane, to a mass of 40 g hanging freely vertically below the pulley. The system was released from rest and the body descended a distance of 686 cm down the plane in the first 2 seconds of motion. Given that the force exerted on the pulley was g-wt, determine the value of . Take .

**Q6: **

Two bodies and of masses 1.8 kg and 3.6 kg, respectively, are connected to each other by a light string and placed on a smooth plane inclined to the horizontal at an angle whose sine is . Body is connected by a light string passing over a smooth pulley fixed to the top of the plane to a third body of mass 2.7 kg. The system was released from rest and started moving. Determine the tension in the string that connects bodies and on the plane, and find the force exerted on the pulley. Take .

- A ,
- B ,
- C ,
- D ,

**Q7: **

A body of mass 2.4 kg rests on a smooth plane inclined at an angle of to the horizontal. It is connected by a light inextensible string passing over a smooth pulley, fixed at the top of the plane, to another body of mass 1.6 kg hanging freely vertically below the pulley. When the system was released from rest, the two bodies were on the same horizontal level. Then 10 seconds later, the string broke. Determine the time taken for the first body to start moving in the opposite direction after the string broke. Take .

**Q8: **

A body of mass 8.3 kg rests on a smooth plane inclined to the horizontal at an angle whose sine is 0.35. It is connected by a light inextensible string passing over a smooth pulley fixed to the top of the plane to another body of mass 8.3 kg resting on a smooth plane inclined to the horizontal at an angle whose sine is 0.95. The two planes meet at the point where the pulley is fixed. The system was released from rest and 4 seconds later the string was cut. Determine the velocity of the second body when the first came to a momentary rest. Take .

**Q9: **

A body of mass 6.8 kg rests on a smooth plane inclined at an angle of to the horizontal. It is connected by a light inextensible string passing over a smooth pulley fixed to the top of the plane to another body of mass 5.1 kg hanging freely vertically below the pulley. When the system was released from rest, the two bodies were on the same horizontal level. Determine the vertical distance between the two bodies 2 seconds after they started moving. Take .

**Q10: **

A body of mass rests on a smooth plane inclined at an angle of to the horizontal. It is connected, by a light inextensible string passing over a smooth pulley fixed at the top of the plane, to a scale pan of mass 270 g hanging freely vertically below the pulley. Another body of mass was placed in the scale pan, and the system is released from rest. The body of mass accelerates up the line of greatest slope of the plane at 70 cm/s^{2}, and the force exerted on the scale pan by the mass is 780 g-wt. Find and in grams. Take the acceleration due to gravity .

- A ,
- B ,
- C ,
- D ,

**Q11: **

A body of mass 222 g rests on a rough plane inclined to the horizontal at an angle whose tangent is . The body is connected by a light inextensible string passing over a smooth pulley fixed at the top of the plane to a body of mass 310 g hanging freely vertically below the pulley. Given that the coefficient of friction between the body and the plane is , find the acceleration of the system. Take the acceleration due to gravity .

**Q12: **

A body of mass 411 g rests on a rough plane inclined to the horizontal at an angle whose tangent is . The body is connected, by a light inextensible string that passes over a smooth pulley fixed at the top of the plane, to a bucket of mass 69 g hanging freely below the pulley. If the coefficient of friction between the body and the plane is , find the minimum mass that must be added to the bucket so that the system remains at rest. Take .

**Q13: **

A body of mass 1.8 kg rests on a rough plane inclined to the horizontal at an angle whose sine is . The coefficient of friction between the body and the plane is . The body is connected, by a light inextensible string passing over a pulley fixed at the top of the plane, to another body of mass 2.4 kg hanging freely vertically below the pulley. The system was released from rest and, after the body had moved 9.8 m up the plane, the string broke. Find the distance covered by the body on the plane before it came to rest. Take the acceleration due to gravity .

**Q14: **

A body of mass 400 g rests on a rough plane inclined to the horizontal at an angle whose tangent is . The body is connected, by a light inextensible string that passes over a smooth pulley fixed at the top of the plane, to a mass of hanging freely vertically below the pulley. Given that the smallest value of necessary to keep the body at rest is 146 g, find the coefficient of friction between the body and the plane. Take the acceleration due to gravity .

**Q15: **

A body of mass 240 g rests on a rough plane inclined to the horizontal at an angle whose sine is . It is connected, by a light inextensible string passing over a smooth pulley fixed to the top of the plane, to another body of mass 300 g. If the system was released from rest and body descended 196 cm in 3 seconds, find the coefficient of friction between the body and the plane. Take .

- A
- B
- C
- D

**Q16: **

A body of mass 162 g rests on a rough plane inclined to the horizontal at an angle whose tangent is . It is connected, by a light inextensible string passing over a smooth pulley fixed to the top of the plane, to another body of mass 181 g hanging freely vertically below the pulley. The coefficient of friction between the first body and the plane is . Determine the distance covered by the system in the first 7 seconds of its movement, given that the bodies were released from rest. Take .

**Q17: **

A body of mass 20 g was resting on a rough plane inclined at to the horizontal. The body was connected to the end of a string passing through a pulley fixed at the top of the plane. At the other end of the string, a body of mass 50 g was hanging freely. Given that the coefficient of friction is , find the time taken for the first body to move a distance of 3.2 m on the plane and its velocity after moving this distance. Take .

- A ,
- B ,
- C ,
- D ,

**Q18: **

A body of mass kg is resting on a smooth plane inclined at to the horizontal. The body is attached by a light inextensible string which passes over a smooth pulley, fixed at the top of the slope, to another body of mass 4 kg lying on a rough horizontal plane. When , the system is in limiting equilibrium and on the point of moving. If , find the vertical distance this body will descend as it moves down the slope over the course of 6 seconds. Take .

- A m
- B m
- C 21 m
- D m

**Q19: **

A body of mass 20 g rests on a rough horizontal plane. It is connected, by a light inextensible string passing over a pulley fixed at the edge of the plane, to another body of mass 180 g resting on a rough plane inclined at an angle of to the horizontal. The two planes meet at the point where the pulley is fixed. The coefficient of friction between the first body and the horizontal plane is , and, between the second body and the inclined plane, it is . Determine the acceleration of the system. Take .

**Q20: **

A mass of 351 g, resting on a rough inclined plane, was connected by a light inextensible string to another mass of 221 g resting on a smooth inclined plane. The angles of inclination of the two planes were and respectively, where and . The connecting string passed over a smooth pulley that was fixed at the apex of the two planes. Given that the first mass was on the point of moving down the plane, determine the coefficient of friction between this body and the rough plane. Take .

- A
- B
- C
- D

**Q21: **

Two bodies of masses 58 g and 124 g are attached to the ends of a light inextensible string passing over a smooth pulley. The first body is on a rough horizontal plane, and the second body on a rough plane inclined at an angle of to the horizontal. The pulley is attached to the line of intersection of the two planes such that the sections of string on each side of the pulley are perpendicular to that line. The coefficient of friction between the first body and the horizontal plane is , and, between the second body and the inclined plane, it is . Given that the string broke 4 seconds after the bodies started moving, determine the distance covered by the body on the horizontal plane before it came to rest. Take the acceleration due to gravity .

**Q22: **

Two smooth inclined planes of the same length are connected at their apex such that each plane is inclined at to the horizontal. A body of mass 10 kg is placed on one of the planes. It is connected, by a light inextensible string passing over a smooth pulley fixed at the top of the planes, to another body of mass 2.5 kg resting on the other plane. The system was released from rest when the two bodies were on the same horizontal level. Determine the force exerted on the pulley and the vertical distance between the two bodies 4 seconds after they started moving. Take .

- A ,
- B ,
- C ,
- D ,

**Q23: **

Two smooth inclined planes meet at a common apex. The first plane is inclined at an angle to the horizontal, and the second is inclined to the horizontal at an angle whose sine is 0.5. A body of mass grams was placed on the first plane. It was connected, by a light inextensible string passing over a smooth pulley fixed at the apex of the two planes, to another body of mass grams resting on the other plane. When the system was released from rest, the two bodies were on the same horizontal level and, 1 second later, the vertical distance between the two bodies was 14 cm. Given that was moving downwards, and upwards, find the ratio between the masses. Take the acceleration due to gravity .

- A
- B
- C
- D

**Q24: **

Three planes are connected: one of them is horizontal and the other two are inclined at an angle of to the horizontal such that the vertical cross section of the three planes forms an isosceles trapezium. A body of mass 14 kg rests on the horizontal plane. It is connected by two light inextensible strings passing over two smooth pulleys, fixed at either end of the horizontal plane, at the points of intersection with the inclined planes, to another two bodies and , of masses 8 kg and 13 kg, respectively, resting on their respective inclined planes. The system was released from rest, and, 6 seconds later, the string connecting bodies and was cut. Determine the velocity of the system formed of bodies and when body came momentarily to rest. Take .

- A m/s
- B m/s
- C m/s
- D m/s

**Q25: **

Two bodies of masses 84 g and 116 g are attached to the ends of a light inextensible string passing over a smooth pulley. The first body is on a rough horizontal plane, and the second on a rough plane inclined at an angle of to the horizontal. The pulley is attached to the line of intersection of the two planes such that the sections of string on each side of the pulley are perpendicular to that line. The coefficient of friction between the first body and the horizontal plane is , and, between the second body and the inclined plane, it is . Determine the magnitude of the tension in the string. Take .