Mechanics: Moments - Worksheets, Questions and Revision

13 original exam-style questions - 5 pages of questions with a full mark scheme - free printable PDF.

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A-Level · Mechanics

M3 Mechanics: Moments

EDEXCEL 9MA0 · Calculator allowed · about 150 minutes
Total Marks
Name: _______________________________    Date: ____ / ____ / ______
Answer ALL questions. Show all your working.
1
A door is fitted with a single hinge mechanism. A cleaner pushes on the door with a force of 150 N applied at right angles to the door, at a point 0.8 m from the hinge.
(a)Calculate the moment of the 150 N force about the hinge, stating the appropriate units.(2)
(b)The hinge mechanism is rated to safely withstand a maximum moment of 100 N m before it risks damage. Determine, with a reason, whether the hinge is at risk of damage from this push.(2)
(Total for Question 1 is 4 marks)
2
A mechanic applies a force of 60 N to the end of a spanner of length 0.24 m, at an angle of 70 degrees to the spanner, in order to loosen a nut. The nut is modelled as a fixed pivot at the other end of the spanner.
(a)Calculate the perpendicular distance from the pivot to the line of action of the force, giving your answer to 3 significant figures.(2)
(b)Hence calculate the moment of the force about the pivot, stating the appropriate units, giving your answer to 3 significant figures.(2)
(c)State, with a reason, whether the moment calculated in part (b) would increase or decrease if the angle between the force and the spanner were reduced to 40 degrees, with the magnitude of the force unchanged.(2)
(Total for Question 2 is 6 marks)
3
A uniform beam AB has length 5 m and weight 240 N. The beam rests horizontally in equilibrium on two smooth supports, one at C, 0.8 m from A, and one at D, 1.2 m from B.
(a)By taking moments about C, show that the reaction at D, RD, is 136 N.(4)
(b)Find the reaction at C, RC.(2)
(c)State two assumptions made about the beam and its supports in this model, and explain briefly how each assumption is used in the calculation.(3)
(Total for Question 3 is 9 marks)
4
A non-uniform rod AB has length 3.6 m. The rod is held in equilibrium in a horizontal position by two vertical strings, one attached at A and the other attached at a point 2.4 m from A. The tension in the string at A is 45 N and the tension in the other string is 75 N.
(a)Find the weight of the rod.(2)
(b)By taking moments about A, find the distance from A to the centre of mass of the rod.(4)
(c)State, without further calculation, whether the centre of mass lies closer to A or to the point where the 75 N string is attached. Justify your answer.(2)
(Total for Question 4 is 8 marks)
5
A see-saw is modelled as a uniform rod PQ of length 4 m and weight 150 N, resting on a smooth pivot at its midpoint. Chandrika, of weight 320 N, sits at P. Tafari, of weight 400 N, sits on the see-saw between the pivot and Q so that the see-saw is horizontal and in equilibrium.
(a)Explain why the weight of the see-saw does not appear in the moments equation about the pivot.(1)
(b)Find the distance of Tafari from the pivot.(4)
(c)A third child, Nia, of weight 250 N, now also sits at Q itself, 2 m from the pivot, while Chandrika and Tafari stay in their original positions. Find the vertical force that must now be applied at P to restore equilibrium, stating whether it acts upward or downward.(5)
(Total for Question 5 is 10 marks)
6
A uniform rod AB has length 1.5 m and weight 60 N. The rod is smoothly hinged to a vertical wall at A and held in a horizontal position by a light inextensible string attached to the rod at B, making an angle of 55 degrees with the rod. A shop sign of weight 45 N hangs from a hook at B.
(a)By taking moments about A, find the tension in the string, giving your answer to 3 significant figures.(4)
(b)Find the magnitude of the vertical component of the force exerted on the rod by the hinge at A.(3)
(c)Given additionally that the horizontal component of the hinge force has magnitude 52.5 N, find the angle that the resultant hinge force makes with the rod.(3)
(Total for Question 6 is 10 marks)
7
A uniform beam AB has length 8 m and weight 260 N. The beam is held horizontally in equilibrium by two vertical cables attached to the beam at points C and D, where AC = 1.5 m and AD = 6.5 m. A load of weight 140 N is attached to the beam at point E, where AE = 3 m.
(a)By taking moments about C, show that the tension in the cable at D, TD, is 172 N.(4)
(b)Find the tension in the cable at C, TC.(2)
(c)The load is now moved to the midpoint of the beam, with everything else unchanged. Find the new tensions in the two cables.(5)
(Total for Question 7 is 11 marks)
8
A uniform rod AB has length 2.4 m and weight 90 N. The rod is smoothly hinged to a vertical wall at A and held in a horizontal position by a light inextensible string attached to the rod at B and to a point on the wall above A, the string making an angle of 35 degrees with the rod.
(a)By taking moments about A, find the tension in the string, giving your answer to 3 significant figures.(4)
(b)Find the magnitude of the horizontal and vertical components of the force exerted on the rod by the hinge at A, stating the direction in which each acts.(5)
(c)Hence find the magnitude and direction of the resultant force exerted by the hinge on the rod.(3)
(Total for Question 8 is 12 marks)
9
A uniform rod AB has length 1.8 m and weight 40 N. The rod rests horizontally on a smooth pivot at its midpoint M. A mass hanging from a string attached at A exerts a force of 25 N vertically downward on the rod. A force of magnitude P is applied to the rod at B, at an angle of 50 degrees above the horizontal, directed so that its vertical component acts upward, such that the rod remains horizontal and in equilibrium.
(a)Explain why the weight of the rod produces no moment about M.(1)
(b)By taking moments about M, find the value of P, giving your answer to 3 significant figures.(4)
(c)Find the magnitude of the force exerted on the rod by the pivot at M, giving your answer to 3 significant figures.(5)
(Total for Question 9 is 10 marks)
10
A uniform plank AB has length 6 m and weight 200 N. The plank rests horizontally on two smooth supports, one at C, 1 m from A, and one at D, 1 m from B. A person of weight 800 N walks slowly along the plank from A to B. Let x metres be the distance of the person from A, where 0 ≤ x ≤ 6. The plank remains in equilibrium as long as both support reactions are greater than or equal to zero; it tips about a support if that support's reaction would otherwise need to be negative.
(a)By taking moments about C, show that, while the person is between A and B, the reaction at D is given by RD = 200x - 100 (in newtons).(4)
(b)Find, in terms of x, an expression for RC, the reaction at support C.(2)
(c)Find the range of values of x for which the plank remains in equilibrium without tipping.(4)
(d)Hence state the length of plank over which the person can walk while the plank remains in equilibrium.(1)
(Total for Question 10 is 11 marks)
11
A uniform ladder AB has length 5 m and weight 200 N. The ladder rests with end A on rough horizontal ground and end B against a smooth vertical wall, making an angle of 68 degrees with the horizontal ground. The ladder is on the point of slipping.
(a)State the four forces acting on the ladder, including the direction in which each acts.(4)
(b)By resolving vertically, find the normal reaction at A, NA.(2)
(c)By taking moments about A, find the normal reaction at the wall, NB, giving your answer to 3 significant figures.(4)
(d)Given that the ladder is on the point of slipping, find the coefficient of friction between the ladder and the ground, giving your answer to 3 significant figures.(4)
(Total for Question 11 is 14 marks)
12
A uniform ladder AB has length 6 m and weight 250 N. The foot A rests on rough horizontal ground, where the coefficient of friction between the ladder and the ground is 0.3, and the top B rests against a smooth vertical wall. The ladder makes an angle of 60 degrees with the ground. Priya, of weight 700 N, climbs up the ladder. Let d metres be the distance she has climbed up the ladder, measured from A.
(a)Write down the normal reaction at A, NA.(1)
(b)By taking moments about A, show that the normal reaction at the wall, NB, is given by NB = (750 + 700d) / (6 tan(60 degrees)) newtons.(4)
(c)Find the maximum distance d that Priya can climb up the ladder before it slips.(5)
(Total for Question 12 is 10 marks)
13
A uniform rod AC of length 2L and weight W is smoothly hinged to a vertical wall at A. The rod is held horizontal by a light inextensible string attached to the rod at C and to a point on the wall above A, the string making an angle θ with the rod.
(a)By taking moments about A, show that the tension in the string is T = W / (2 sin(θ)).(4)
(b)Show that the magnitude of the force exerted on the rod by the hinge at A is also equal to T.(5)
(c)Given that θ = 40 degrees and W = 150 N, find the magnitude of the force exerted by the hinge on the rod, giving your answer to 3 significant figures.(3)
(Total for Question 13 is 12 marks)
Mark scheme · M3 Mechanics: Moments

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