Further Mechanics: Work, Energy and Power - Worksheets, Questions and Revision

13 original exam-style questions - 4 pages of questions with a full mark scheme - free printable PDF.

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A-Level · Further Mechanics 2 (Work, Energy and Power)

FP.FM2 Further Mechanics: Work, Energy and Power

AQA 7367 · Calculator allowed · about 110 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.

Formulae and modelling assumptions used in this worksheet

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Unless a question states otherwise, take g = 9.8 m s^-2 and model each object as a particle. Strings and ropes are light and inextensible, and any pulleys or engines are treated as smooth or ideal unless stated. Key results used across this worksheet: work done by a constant force F acting through a displacement s at angle theta to the direction of motion is W = F*s*cos(theta); kinetic energy is KE = (1/2)*m*v^2; gravitational potential energy relative to a reference level is GPE = m*g*h; the work-energy principle states that the total work done by all the forces acting on a particle equals the change in its kinetic energy; power is the rate of doing work, so for a force F moving its point of application at speed v in the direction of F, P = F*v, and the work done over a time interval is W = integral of P with respect to t; for a force that varies with displacement, F(x), the work done as the point of application moves from x = a to x = b is W = integral from a to b of F(x) dx.

1
A courier, Priya, pulls a parcel of mass 4 kg across a horizontal warehouse floor using a rope inclined at 30 degrees to the horizontal. The tension in the rope is a constant 20 N, and a constant resistance to motion of 8 N acts on the parcel as it slides. The parcel starts from rest.
Figure (to be drawn): Diagram: parcel on a horizontal floor; a rope from the parcel rises at 30 degrees above the horizontal to Priya's hand, with tension T = 20 N marked along the rope; a resistance force R = 8 N is shown acting horizontally, opposing the direction of motion.
(a)Given that the parcel moves a distance of 5 m across the floor, calculate the work done by the tension in the rope.(2)
(b)Using the work-energy principle, find the speed of the parcel after it has moved this 5 m distance.(4)
(Total for Question 1 is 6 marks)
2
A resultant force of constant magnitude 30 N acts on a body of mass 5 kg, in the direction of its motion, while the body is initially moving at 2 m/s. The force continues to act as the body moves a further distance of 10 m in a straight line.
(a)Find the speed of the body after it has moved the 10 m distance.(4)
(Total for Question 2 is 4 marks)
3
A small block of mass 1.5 kg is released from rest at the top of a smooth incline of length 6 m, inclined at 20 degrees to the horizontal, and slides down to the bottom of the incline.
Figure (to be drawn): Diagram: block at the top of a smooth incline of length 6 m, inclined at 20 degrees to the horizontal; weight mg shown vertically downward, normal reaction N perpendicular to the incline surface, with the block released from rest at the top.
(a)Use energy methods to find the speed of the block at the bottom of the incline.(3)
(b)Show, by resolving forces along the incline and using a suvat equation, that the same speed at the bottom is obtained as in part (a).(3)
(Total for Question 3 is 6 marks)
4
A cyclist, Owen, and his cycle have a combined mass of 80 kg. Cycling along a horizontal road at a constant speed of 6 m/s, Owen experiences a constant resistance to motion of 40 N.
Figure (to be drawn): Diagram: (part a) cyclist on a horizontal road with driving force F and resistance R = 40 N shown in opposite horizontal directions; (part b) the same cyclist on a road inclined at 5 degrees to the horizontal, with the weight component mg*sin(5 degrees) acting down the slope alongside the resistance R.
(a)Calculate the power that Owen develops while cycling at this constant speed on the horizontal road.(2)
(b)Owen then rides up a hill inclined at 5 degrees to the horizontal, continuing to develop the same constant power as in part (a) and experiencing the same resistance of 40 N. Find the new constant speed at which Owen travels up the hill.(5)
(Total for Question 4 is 7 marks)
5
A car of mass 900 kg experiences a constant resistance to motion of magnitude 500 N, and its engine works at a constant power of 20 kW.
Figure (to be drawn): Diagram: car on a road inclined at angle α to the horizontal, where sin(α) = 1/14; forces shown are the driving force F (up the slope), resistance R = 500 N (down the slope) and the weight component m*g*sin(α) (down the slope).
(a)Find the maximum speed of the car on a horizontal road.(3)
(b)The car now travels up a straight road inclined at angle α to the horizontal, where sin(α) = 1/14. At an instant when the car's speed is 25 m/s, the engine still works at the same power of 20 kW and the resistance to motion is still 500 N. Find the acceleration of the car at this instant.(6)
(Total for Question 5 is 9 marks)
6
A warehouse worker, Tomasz, pushes a crate of mass 25 kg up a rough plane inclined at 15 degrees to the horizontal, using a constant force of 200 N acting parallel to the plane. The coefficient of friction between the crate and the plane is 0.3. The crate starts from rest and moves a distance of 4 m up the plane.
Figure (to be drawn): Diagram: crate on a rough incline of angle 15 degrees to the horizontal; applied force P = 200 N acts up the slope, friction F acts down the slope (opposing motion), weight m*g acts vertically down, and normal reaction N acts perpendicular to the slope.
(a)Show that the frictional force acting on the crate has magnitude awrt 71.0 N.(3)
(b)Use the work-energy principle to find the speed of the crate after it has moved the 4 m distance up the plane.(5)
(Total for Question 6 is 8 marks)
7
A particle of mass 0.5 kg is projected up a line of greatest slope of a rough plane inclined at 20 degrees to the horizontal, with initial speed 8 m/s. The coefficient of friction between the particle and the plane is 0.25.
Figure (to be drawn): Diagram: particle on a rough incline of angle 20 degrees to the horizontal, projected up the line of greatest slope with initial speed 8 m/s; weight component m*g*sin(20 degrees) and friction force both act down the slope, opposing the upward motion.
(a)Use the work-energy principle to find the distance the particle travels up the plane before it comes to instantaneous rest.(5)
(b)Determine, with justification, whether the particle subsequently slides back down the plane.(3)
(Total for Question 7 is 8 marks)
8
A particle of mass 3 kg moves in a straight line along the x-axis under the action of a single force F newtons, where F = (18 - 2x) and x is the displacement in metres from the particle's starting point, for 0 ≤ x ≤ 9. The particle starts at x = 0 with speed 2 m/s.
(a)Find the work done by F as the particle moves from x = 0 to x = 5.(4)
(b)Find the speed of the particle when x = 5.(3)
(c)Explain, without further calculation, why the particle's speed is increasing throughout its motion from x = 0 to x = 9.(2)
(Total for Question 8 is 9 marks)
9
A go-kart of mass 120 kg moves from rest along a smooth horizontal track. At time t seconds after starting (0 ≤ t ≤ 8), the power developed by its engine is P = (300 + 40t) watts.
(a)Find the total work done by the engine during the first 8 seconds of motion.(3)
(b)Hence find the speed of the go-kart at t = 8 seconds.(3)
(Total for Question 9 is 6 marks)
10
A sledge of mass 40 kg is pulled up a straight slope inclined at 10 degrees to the horizontal by a rope parallel to the slope, in which the tension is a constant 350 N. Due to the changing surface of the slope, the resistance to motion (excluding gravity) when the sledge is a distance x metres up the slope (0 ≤ x ≤ 12) is modelled by R = (15 + 2x) newtons. The sledge starts from rest at the bottom of the slope.
Figure (to be drawn): Diagram: sledge on a straight slope inclined at 10 degrees to the horizontal, pulled up the slope by a rope of constant tension T = 350 N parallel to the slope; the variable resistance force R(x) = (15 + 2x) N and the weight component m*g*sin(10 degrees) both act down the slope, opposing motion.
(a)Find the work done against the variable resistance as the sledge moves from x = 0 to x = 12.(3)
(b)Using the work-energy principle, find the speed of the sledge when x = 12.(6)
(Total for Question 10 is 9 marks)
11
This question concerns the work-energy principle for a particle moving under a system of forces.
(a)State the work-energy principle.(1)
(b)A particle of mass 2 kg is initially moving at 3 m/s. A resultant force acting on the particle does 60 J of work on it as it moves. Find the final speed of the particle.(3)
(Total for Question 11 is 4 marks)
12
A car of mass 1200 kg experiences a resultant driving force of 900 N while moving at a constant speed of 18 m/s along a horizontal road.
(a)Calculate the power developed by the engine.(3)
(Total for Question 12 is 3 marks)
13
A skier, Aisha, of mass 65 kg starts from rest at the top of a slope inclined at 12 degrees to the horizontal and skis down a distance of 150 m measured along the slope. The resistance to motion, from friction and air resistance combined, has constant magnitude 40 N.
Figure (to be drawn): Diagram: skier at the top of a slope inclined at 12 degrees to the horizontal, moving down the line of greatest slope; weight component m*g*sin(12 degrees) acts down the slope (in the direction of motion) and a resistance force R = 40 N acts up the slope, opposing motion.
(a)Use the work-energy principle to find Aisha's speed at the bottom of the slope.(5)
(b)State one physical factor, other than air resistance and friction, that this model does not take into account and that could affect the accuracy of the calculated speed.(1)
(Total for Question 13 is 6 marks)
Mark scheme · FP.FM2 Further Mechanics: Work, Energy and Power

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13