Worksheet: Motion of a Body on a Rough Inclined Plane

In this worksheet, we will practice solving problems on a particle moving on a rough inclined plane by resolving the forces parallel and perpendicular to the plane.

Q1:

A car of mass π‘š tons was initially at rest on a hill inclined at an angle πœƒ to the horizontal, where s i n πœƒ = 1 2 . After 100 seconds, its velocity was 21 m/s. Calculate the resistance per ton of the car’s mass. Take 𝑔 = 9 . 8 / m s  .

Q2:

A body was placed on the top of a rough inclined plane of length 259 cm and height 84 cm. The body started sliding down the plane. Given that the coefficient of friction was 0.29 , find the acceleration of the body. Take 𝑔 = 9 . 8 / m s  .

Q3:

A small wooden box of mass 11 kg was placed at the top of a rough inclined plane of length 2.25 m and height 1.8 m. Given that it slid down the plane in 1 second, find the magnitude of its acceleration π‘Ž and the magnitude of the friction 𝐹 between the box and the plane. Take the acceleration due to gravity 𝑔 = 9 . 8 / m s  .

  • A π‘Ž = 2 . 2 5 / m s  , 𝐹 = 6 4 . 6 8 N
  • B π‘Ž = 2 . 2 5 / m s  , 𝐹 = 6 1 . 4 9 N
  • C π‘Ž = 4 . 5 / m s  , 𝐹 = 6 4 . 6 8 N
  • D π‘Ž = 4 . 5 / m s  , 𝐹 = 3 6 . 7 4 N

Q4:

In a factory, boxes are transferred from one area to another via a rough inclined slope of length 13 m and height 12 m. The boxes are released from rest at the top of the slope and left to slide down freely. Given that the coefficient of friction between the plane and a box is 0.27 , find the speed of a box when it reaches the bottom of the slope rounded to two decimal places. Take the acceleration due to gravity 𝑔 = 9 . 8 / m s  .

Q5:

A body of mass 30 kg was projected at 12 m/s along the line of greatest slope of a plane inclined at 3 0 ∘ to the horizontal. Given that the resistance of the plane to the movement of the body was 3 N, how long did it take for the body to come to rest? Consider the acceleration due to gravity to be 9.8 m/s2.

Q6:

A body was being projected up a rough inclined plane of length 300 cm whose heighest point is 280 cm from the ground. If the coefficient of friction between the body and the plane was 0.41 , find, to the nearest two decimal places, the minimum speed at which the body must be projected to reach the top. Give your answer in centimeters per second. Take the acceleration due to gravity to be 𝑔 = 9 . 8 / m s  .

Q7:

A body is projected at 14.28 m/s up the line of greatest slope of a plane inclined to the horizontal at an angle whose tangent is √ 2 4 . If the coefficient of friction between the plane and the body is 3 √ 2 5 , what is the maximum distance that the body can travel up the plane? (Take 𝑔 = 9 . 8 / m s  ).

Q8:

A body of mass 74 kg was projected at 8.5 m/s along the line of greatest slope up a plane inclined at 3 0 ∘ to the horizontal. Given that the resistance of the plane to its motion was 7.4 N, find the distance the body travelled until it came to rest. Take 𝑔 = 9 . 8 / m s  .

Q9:

A body, starting at rest, slides down a rough plane inclined at an angle of 4 5 ∘ to the horizontal. The coefficient of friction between the body and the plane is 3 4 . Let 𝑑  be the timerequired to traverse a certain distance down the slope and 𝑑  be the time required for the same body to travel the same distance distance down a smooth slope at the same angle of inclination. Express 𝑑  in terms of 𝑑  .

  • A 𝑑 = 4 𝑑  
  • B 𝑑 = 𝑑  
  • C 𝑑 = √ 2 𝑑  
  • D 𝑑 = 2 𝑑  

Q10:

A car was moving down a hill inclined at an angle πœƒ to the horizontal, where s i n πœƒ = 4 7 5 . When its engine was off, it moved at a constant speed. If the same car was moving up the same slope at 2.8 m/s and its engine cut out, how far would it move before it came to rest? Assume that the magnitude of the resistance to its movement is the same while both ascending and descending. Take 𝑔 = 9 . 8 / m s  .

Q11:

A body was projected at 53.9 m/s up a rough plane inclined at an angle of 3 0 ∘ to the horizontal. If the body came to rest 5 seconds after being projected, determine the coefficient of friction of the plane πœ‡ , and determine whether the body will return to the point from which it was projected or not. Take 𝑔 = 9 . 8 / m s  .

  • A πœ‡ = 1 6 √ 3 1 5 , will not return
  • B πœ‡ = 2 √ 3 5 , will return
  • C πœ‡ = 1 6 √ 3 1 5 , will return
  • D πœ‡ = 2 √ 3 5 , will not return

Q12:

A locomotive of mass 110 metric tons has an engine which generates a force of 216 kN. The locomotive is pulling a number of carriages up a section of track inclined at an angle πœƒ to the horizontal, where s i n πœƒ = 1 1 0 . Given that the mass of each carriage is 4 metric tons, the resistance to the locomotive’s movement is 30 kg-wt for each tonne of the locomotive’s mass, and the locomotive is accelerating at 16.6 cm/s2, determine the number of the carriages the locomotive is pulling. Take 𝑔 = 9 . 8 / m s  .

Q13:

A body weighing 3 0 0 √ 3 N was placed on a rough plane inclined at an angle of 6 0 ∘ to the horizontal. The coefficient of static friction between the body and the plane was √ 3 5 , and the coefficient of kinetic friction was √ 3 6 . Find the force 𝐹  that causes the body to be on the point of moving up the plane, and then determine the minimum force 𝐹  that will maintain the motion of the body if it is already moving up the plane.

  • A 𝐹 = 3 6 0  N , 𝐹 = 3 7 5  N
  • B 𝐹 = 5 2 5  N , 𝐹 = 5 4 0  N
  • C 𝐹 = 3 7 5  N , 𝐹 = 3 6 0  N
  • D 𝐹 = 5 4 0  N , 𝐹 = 5 2 5  N

Q14:

A body of mass π‘š kg was placed on a plane inclined at 4 5 ∘ to the horizontal. A force of magnitude 3 9 2 √ 2 N was acting on the body along the line of greatest slope up the plane. As a result, the body accelerated uniformly at π‘Ž m/s2 up the plane. If the magnitude of the force acting on the body is halved while maintaining its original direction, the body will move down the plane at π‘Ž m/s2. Given that the resistance of the plane to the body’s movement is 3 8 √ 2 N in both cases, determine the values of π‘š and π‘Ž , rounding the results to the nearest two decimal places. Take 𝑔 = 9 . 8 / m s  .

  • A π‘š = 4 0 . 0 0 k g , π‘Ž = 3 . 2 1 / m s 
  • B π‘š = 6 0 . 0 0 k g , π‘Ž = 3 . 2 1 / m s 
  • C π‘š = 4 0 . 0 0 k g , π‘Ž = 8 . 2 7 / m s 
  • D π‘š = 6 0 . 0 0 k g , π‘Ž = 1 . 4 1 / m s 

Q15:

A body of mass 3 7 √ 3 kg slid down the line of greatest slope of a rough plane inclined at 6 0 ∘ to the horizontal, and onto a rough horizontal plane. It continued to slide along the horizontal plane, initially at the same the speed it left the inclined plane, before coming to rest. Given that it slid the same distance along each of the planes, and that the resistance to its motion was constant, find the magnitude of this resistive force. Consider the acceleration due to gravity to be 9.8 m/s2.

Q16:

A body was placed at the top of a rough inclined plane of length 400 cm and height 240 cm. Given that the body started sliding down the plane and that the coefficient of friction between the plane and the body was 0.63, find the velocity of the body after it had moved 150 cm down the plane. Take the acceleration due to gravity 𝑔 = 9 . 8 / m s  .

Q17:

A body was projected up a rough inclined plane of length 45 m and height 22 m from its bottom. Given that the friction of the plane is 0.4 times the weight of the body, determine the minimum speed at which the body must be projected to reach the top of the plane. Take 𝑔 = 9 . 8 / m s  .

Q18:

A train of mass 270 metric tons is accelerating along a horizontal track at 4.4 cm/s2. Its engine produces a driving force of 2,080 kg-wt. If the train starts to ascend a 1 in 490 incline, find the acceleration of the train given that the resistance does NOT change, and the acceleration due to gravity is 𝑔 = 9 . 8 / m s  .

Q19:

A car weighing 1,920 kg-wt was driving along a straight horizontal road at a constant speed. It reached the top of a ramp inclined at an angle πœƒ to the horizontal, where s i n πœƒ = 1 2 7 . At this point, the driver stopped the engine so the car was left to freely move down the slope. As it moved down the slope, it maintained a constant speed. Given that the resistance of the ramp is 1 9 of the resistance of the horizontal road, calculate what the driving force of the motor was when the car was moving along the horizontal road. Consider the acceleration due to gravity to be 9.8 m/s2.

Q20:

A train of mass 110 metric tons was accelerating at 7.4 cm/s2 up a plane that was inclined at an angle πœƒ to the horizontal, where s i n πœƒ = 1 1 0 0 . Given that the combined magnitude of the air resistance and friction was 4 kg-wt for each tonne of the train’s mass, find the force generated by the train’s engine. Take 𝑔 = 9 . 8 / m s  .

Q21:

A rough plane was inclined at an angle πœƒ to the horizontal, where s i n πœƒ = 5 7 . A body was projected upwards from the bottom of the plane with a velocity of 22.4 m/s in the direction of the line of greatest slope. If the coefficient of the kinetic friction between the body and the plane was √ 6 4 , find the time taken for the body to come to rest from the moment of its projection. Consider the acceleration due to gravity to be 9.8 m/s2.

Q22:

A car of mass 800 kg was descending a plane inclined at an angle πœƒ to the horizontal, where s i n πœƒ = 1 2 5 . The resistance to the movement of the car was 15 kg-wt per tonne of its mass. Given that, starting from rest, the car covered 28 m in 4 seconds, find the force of its motor. Take 𝑔 = 9 . 8 / m s  .

Q23:

A car of mass 4 metric tons was accelerating down a plane inclined to the horizontal at an angle whose sine is 1 3 2 at 86 cm/s2. Given that the resistance of the plane to the car’s motion equals 10 kg-wt per tonne of its mass, find the force generated by the engine of the car. Consider the acceleration due to gravity to be 𝑔 = 9 . 8 / m s  .

Q24:

A locomotive was pulling a train of mass 512 metric tons up a straight track inclined at an angle πœƒ to the horizontal, where s i n πœƒ = 1 2 5 . The pulling force of the locomotive was 32 ton-weight. Given that the total resistance to the movement of the train was 21 kg-wt per metric ton of the train’s mass, find the magnitude of the acceleration of the train. Consider the acceleration due to gravity to be 9.8 m/s2.

Q25:

A locomotive of mass 60 tonnes has an engine with a force equal to 27 tonne weight is pulling a train of carriages, where each carriage weighs 9 tonnes. The locomotive ascended a railway inclined to the horizontal at an angle πœƒ , where s i n πœƒ = 1 8 , and the resistance to its motion is 15 kg-wt per tonne of mass. If the train ascended the railway at an acceleration of 20.3 cm/s2, determine the number of carriages. Consider the acceleration due to gravity to be 9.8 m/s2.

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