Moments – Year 13 A level maths worksheet with answers to download as PDF

Calculate the moment of each force about the point \(O\), stating whether it is clockwise or anticlockwise where this is given.

1. A force of 24 N acts at right angles to a spanner at a distance of 0.35 m from \(O\). [1 mark]
2. A force of 40 N acts on a rod at an angle of \(50^\circ\) to the rod, at a point 0.6 m from \(O\) along the rod. [2 marks]
3. A force of 75 N has a moment of 30 N m about \(O\). Find the perpendicular distance of its line of action from \(O\). [1 mark]

A uniform seesaw has its pivot at its centre. A child of mass 28 kg sits 1.8 m from the pivot.

1. At what distance from the pivot must a child of mass 36 kg sit on the other side to balance the seesaw? [3 marks]
2. A third child of mass 18 kg sits 0.5 m beyond the 36 kg child. Where must the 28 kg child sit now to keep balance? [3 marks]

A uniform beam \(AB\) of length 5 m and mass 30 kg rests on supports at \(A\) and \(B\). A load of mass 15 kg is placed on the beam 2 m from \(A\).

1. Draw a force diagram. [2 marks]
2. By taking moments about \(A\), find the reaction at \(B\). [3 marks]
3. Find the reaction at \(A\). [2 marks]

A uniform rod \(AB\) of length 8 m and mass 16 kg rests horizontally on supports at \(C\) and \(D\), where \(AC=1\) m and \(BD=2\) m.

1. Find \(CD\). [1 mark]
2. Find the reactions at \(C\) and \(D\). [5 marks]

A uniform plank of length 5 m and mass 20 kg rests on two supports, each 0.5 m from an end. A man of mass 50 kg walks along the plank towards one end.

1. Explain why the reaction at the far support is zero when the plank is about to tilt. [1 mark]
2. How far beyond the nearer support can he walk before the plank tilts? [3 marks]
3. Does this allow him to reach the end of the plank? [1 mark]

A uniform rod \(AB\) of length 1.5 m and mass 4 kg is suspended horizontally by two vertical strings attached at points 0.25 m from each end. A particle of mass 6 kg is attached to the rod at a point 0.4 m from \(A\).

1. Find the tension in each string. [6 marks]

A gardener uses a wheelbarrow. The load of mass 45 kg acts 0.5 m horizontally from the axle of the wheel, and the gardener lifts the handles with a vertical force \(F\) at 1.3 m from the axle. The wheelbarrow itself is light.

1. Find \(F\). [3 marks]
2. Find the force the ground exerts on the wheel. [2 marks]
3. The gardener moves the load 0.2 m closer to the wheel. Find the new force \(F\) and comment. [3 marks]

A uniform diving board \(AC\) of length 3 m and mass 24 kg is fixed at \(A\) by a bolt and rests on a support at \(B\), where \(AB=0.8\) m. A diver of mass 60 kg stands at the free end \(C\).

1. Show that the bolt must pull downwards on the board at \(A\). [2 marks]
2. By taking moments about \(B\), find the force exerted by the bolt at \(A\). [4 marks]
3. Find the force exerted by the support at \(B\). [2 marks]

A uniform plank \(AB\) of length 6 m and mass 40 kg lies on two supports, at 1 m from \(A\) and at 1 m from \(B\). A girl of mass 45 kg walks along the plank.

1. The girl stands 2 m from \(A\). Find the reaction at each support. [5 marks]
2. Show that the plank cannot tilt, wherever she stands. [4 marks]

A uniform shelf \(AB\) of length 2.4 m and mass 8 kg is hinged to a wall at \(A\) and supported horizontally by a light cable joining \(B\) to a point on the wall. The cable makes an angle of \(50^\circ\) with the shelf.

1. Show that the tension in the cable is 51.2 N, to 3 significant figures. [4 marks]
2. Find the horizontal and vertical components of the force exerted by the hinge on the shelf. [4 marks]

A plank of length 5 m and mass 20 kg balances horizontally about a pivot 1.8 m from end \(A\) when a child of mass 12 kg sits at \(A\).

1. Find the distance of the centre of mass of the plank from \(A\). [3 marks]
2. The child leaves and a child of mass 8 kg sits at \(B\) instead. Find the distance from \(A\) at which the pivot must now be placed for the plank to balance. [3 marks]

A uniform metre rule has mass 120 g. It balances horizontally on a pivot placed at the 40 cm mark when a small mass \(m\) hangs at the 10 cm mark.

1. Find \(m\). [3 marks]
2. The mass is moved to the 0 cm mark. Find the position of the pivot for balance. [3 marks]

A uniform steel girder \(AB\) of length 8 m and mass 600 kg lies on rough horizontal ground. A crane lifts end \(B\) using a cable that makes \(20^\circ\) with the vertical. The girder is on the point of leaving the ground at \(B\) while end \(A\) stays on the ground.

1. Explain why the vertical component of the cable tension equals half the weight of the girder. [2 marks]
2. Find the tension in the cable. [3 marks]
3. Find the horizontal component of the cable tension, and state what force prevents the girder from sliding. [2 marks]

A uniform fishing rod of length 3 m and mass 0.8 kg is held horizontally by an angler. One hand is at the end \(A\) and the other is 0.3 m from \(A\). A fish pulls vertically downwards with a force of 10 N at the far end \(B\).

1. Find the vertical force exerted by the hand 0.3 m from \(A\). [3 marks]
2. Find the vertical force exerted by the hand at \(A\) and state its direction. [3 marks]

A uniform scaffold plank \(AB\) of length 4.8 m and mass 30 kg rests horizontally on supports at \(C\) and \(D\) where \(AC=0.6\) m and \(AD=4.2\) m. A worker of mass 70 kg stands 1.0 m from \(A\) and a tool box of mass 20 kg rests 3.5 m from \(A\).

1. Find the reactions at \(C\) and \(D\). [5 marks]
2. The worker walks to end \(B\). Find the new reactions and show that the plank does not tilt. [5 marks]
3. State two modelling assumptions. [2 marks]

A uniform rod \(AB\) of length 2 m and mass 10 kg is hinged to a vertical wall at \(A\) and held horizontally by a light cable from \(B\) to the wall which makes an angle of \(40^\circ\) with the rod. A mass of 8 kg hangs from the rod at a point 1.5 m from \(A\).

1. Show that the tension in the cable is 168 N, to 3 significant figures. [4 marks]
2. Find the horizontal and vertical components of the force exerted on the rod by the hinge. [4 marks]
3. Find the magnitude and direction of the hinge force. [3 marks]
4. The cable is replaced by a longer one so that it makes \(25^\circ\) with the rod. Find the new tension and comment. [3 marks]

A uniform rod of mass 6 kg and length 2 m carries four small masses, as shown in the table. The distances are measured from end \(A\).

1. Find the distance from \(A\) of the point at which the loaded rod balances. [4 marks]
2. The rod is placed on a pivot at that point. Find the reaction at the pivot. [1 mark]
3. The 4 kg mass is removed. Find how far the pivot must move, and in which direction, to balance. [4 marks]

A plank of length 4 m rests horizontally on two bathroom scales, one under each end. The scales read 180 N and 220 N, at \(A\) and \(B\) respectively. The plank is non-uniform.

1. Find the weight of the plank and the distance of its centre of mass from \(A\). [4 marks]
2. A person of mass 70 kg now stands on the plank 1 m from \(A\). Find the new reading on each scale. [5 marks]
3. The scales can each read up to 800 N. Find the greatest distance from \(A\) at which the person can stand without overloading scale \(B\). [3 marks]

A uniform rectangular crate has width 0.6 m, height 1.2 m and mass 50 kg. It stands on a rough floor with coefficient of friction 0.4. A horizontal force \(P\) is applied at height 1.0 m above the floor, pushing the crate along the floor.

1. Find the force needed to make the crate slide. [3 marks]
2. Find the force needed to make the crate tip about its lower edge. [4 marks]
3. Which happens first? [1 mark]
4. At what height should the force be applied so that it is certain to slide rather than tip? [3 marks]

A uniform plank of length \(2L\) and mass \(M\) rests on two supports, each at a distance \(d\) from an end. A person of mass \(m\) walks towards one end.

1. Show that the plank tilts when the person is a distance \(x=\dfrac{M(L-d)}{m}\) beyond the nearer support. [3 marks]
2. Hence show that the person can walk to the end without tilting if and only if \(d>\dfrac{ML}{M+m}\). [3 marks]
3. Check your result with \(M=40\), \(m=45\), \(L=3\), \(d=1\). [2 marks]

A uniform rod \(AB\) of weight \(W\) and length \(2a\) is hinged at \(A\) and held horizontally by a light string attached at \(B\) making angle \(\theta\) with the rod.

1. Show that the tension is \(T=\dfrac{W}{2\sin\theta}\). [3 marks]
2. For which value of \(\theta\) is the tension a minimum, and what is it? [2 marks]
3. The string can bear 80 N and \(W=100\) N. Find the smallest value of \(\theta\). [3 marks]