Worksheet: Equilibrium of a Rigid Body under Parallel Forces

In this worksheet, we will practice solving problems about the equilibrium of a body under the effect of parallel coplanar forces.

Q1:

A uniform rod 𝐴 𝐡 having a weight of 64 N and a length of 168 cm is resting horizontally on two identical supports at its ends. A weight of magnitude 56 N is suspended at a point on the rod that is π‘₯ cm away from 𝐴 . If the magnitude of the reaction at 𝐴 is double that at 𝐡 , determine the value of π‘₯ .

Q2:

A uniform rod having a weight of 35 N is resting horizontally on two supports 𝐴 and 𝐡 at its ends, where the distance between the supports is 48 cm. If a weight of magnitude 24 N is suspended at a point that is 38 cm away from 𝐴 , determine the reactions of the two supports 𝑅  and 𝑅  .

  • A 𝑅 = 1 7 . 5  N , 𝑅 = 4 1 . 5  N
  • B 𝑅 = 3 6 . 5  N , 𝑅 = 2 2 . 5  N
  • C 𝑅 = 2 2 . 5  N , 𝑅 = 3 6 . 5  N
  • D 𝑅 = 4 1 . 5  N , 𝑅 = 1 7 . 5  N

Q3:

A non-uniform rod 𝐴 𝐡 having a weight of 40 N and a length of 80 cm is suspended vertically from its midpoint by a light string, and it becomes in equilibrium in a horizontal position when a weight of magnitude 29 N is suspended from its end 𝐴 . Determine the distance π‘₯ between the point at which the weight of the rod is acting and end 𝐴 . After removing the weight at 𝐴 , determine the magnitude of the vertical force that would be needed to keep the rod in equilibrium in a horizontal position when it acts at end 𝐡 .

  • A π‘₯ = 1 1 c m , 𝐹 = 2 9 N
  • B π‘₯ = 2 9 c m , 𝐹 = 1 1 N
  • C π‘₯ = 6 9 c m , 𝐹 = 2 9 N
  • D π‘₯ = 1 1 c m , 𝐹 = 1 1 N

Q4:

A uniform rod 𝐴 𝐡 having a length of 1.3 m and weighing 147 N is resting in a horizontal position on two supports, where the support 𝐢 is at the end 𝐴 , and 𝐷 is at a distance π‘₯ from the end 𝐡 . Find the reaction of the support 𝑅  and the distance π‘₯ , given that 𝑅 = 2 5 𝑅   .

  • A 𝑅 = 4 2  N , π‘₯ = 3 9 c m
  • B 𝑅 = 1 0 5  N , π‘₯ = 9 1 c m
  • C 𝑅 = 1 0 5  N , π‘₯ = 3 9 c m
  • D 𝑅 = 4 2  N , π‘₯ = 9 1 c m

Q5:

A uniform rod 𝐴 𝐡 weighs 70 N and has a length of 95 cm. It is suspended from its ends by two vertical strings, where 𝑇  is the tension of the string at 𝐴 , and 𝑇  is the tension of the string at 𝐡 . A weight of 100 N is suspended from the rod, 30 cm away from 𝐴 , and a weight of 93 N is suspended from the rod, 20 cm away from 𝐡 . Determine the values of 𝑇  and 𝑇  .

  • A 𝑇 = 1 4 0  N , 𝑇 = 1 2 3  N
  • B 𝑇 = 4 0 3  N , 𝑇 = 1 4 0  N
  • C 𝑇 = 1 3 6 . 1 6  N , 𝑇 = 1 2 6 . 8 4  N
  • D 𝑇 = 1 2 6 . 8 4  N , 𝑇 = 1 3 6 . 1 6  N
  • E 𝑇 = 1 2 3  N , 𝑇 = 1 4 0  N

Q6:

𝐴 𝐡 is a uniform rod having a length of 111 cm and weighing 78 N. The rod is suspended horizontally from its ends 𝐴 and 𝐡 by two vertical strings. Given that a weight of 111 N is suspended π‘₯ cm away from the end 𝐴 so that the tension at 𝐴 is twice that at 𝐡 , determine the tension at 𝐡 and the value of π‘₯ .

  • A 𝑇 = 1 2 6 N , π‘₯ = 2 4 c m
  • B 𝑇 = 6 3 N , π‘₯ = 2 4 c m
  • C 𝑇 = 6 3 N , π‘₯ = 8 7 c m
  • D 𝑇 = 1 2 6 N , π‘₯ = 8 7 c m

Q7:

𝐴 𝐡 is a uniform rod having a length of 78 cm and weighing 155 N. The rod is resting horizontally on two supports, 𝐴 and 𝐢 , where 𝐢 is 13 cm away from 𝐡 . Determine the minimum weight 𝑀 to be suspended at 𝐡 so that there is no pressure at 𝐴 , and find out the pressure on 𝐢 at that instant.

  • A 𝑀 = 3 1 0 N , 𝑃 = 1 5 5 N
  • B 𝑀 = 3 1 0 N , 𝑃 = 4 6 5 N
  • C 𝑀 = 7 7 . 5 N , 𝑃 = 7 7 . 5 N
  • D 𝑀 = 7 7 . 5 N , 𝑃 = 2 3 2 . 5 N

Q8:

𝐴 𝐡 is a uniform rod with length 48 cm and weight 20 kg-wt. It rests in a horizontal position on two supports, 𝐢 and 𝐷 , that are 6 cm and 12 cm away from 𝐴 and 𝐡 , respectively. A weight of magnitude 26 kg-wt is suspended from the rod at a point 12 cm away from 𝐴 . Another weight of 16 kg-wt is suspended from the rod, 18 cm away from 𝐡 . Calculate the size of the reaction forces, 𝑅  and 𝑅  , at 𝐢 and 𝐷 respectively.

  • A 𝑅 = 2 8  k g - w t , 𝑅 = 3 4  k g - w t
  • B 𝑅 = 9 6  k g - w t , 𝑅 = 3 4  k g - w t
  • C 𝑅 = 7 2  k g - w t , 𝑅 = 3 0  k g - w t
  • D 𝑅 = 3 2  k g - w t , 𝑅 = 3 0  k g - w t

Q9:

A uniform rod 𝐴 𝐡 having a weight of 30 N and a length of 190 cm is resting horizontally on two supports 𝐢 and 𝐷 , where 𝐢 is near to 𝐡 , and 𝐷 is near to 𝐴 . If the pressure on 𝐢 is double that on 𝐷 , where the distance between them is 66 cm, determine the lengths of 𝐢 𝐡 and 𝐴 𝐷 .

  • A 𝐢 𝐡 = 1 1 7 c m , 𝐴 𝐷 = 5 1 c m
  • B 𝐢 𝐡 = 7 3 c m , 𝐴 𝐷 = 5 1 c m
  • C 𝐢 𝐡 = 2 2 c m , 𝐴 𝐷 = 4 4 c m
  • D 𝐢 𝐡 = 7 3 c m , 𝐴 𝐷 = 7 c m

Q10:

A uniform rod having a length of 114 cm and a weight of 66 N is suspended horizontally by means of two vertical strings at its ends. The greatest tension each string can handle is 87 N. If a weight of 76 N is to be attached to the rod, find the minimum distance at which it can be hung away from the string that has the maximum tension.

Q11:

The length of a rod 𝐴 𝐡 is 111 cm, and its weight is 95 newtons, which is acting at its midpoint. The rod is resting horizontally on two supports, where one of them is at end 𝐴 , and the other is at a point 𝐢 which is 30 cm away from 𝐡 . A weight of 71 newtons is suspended from the rod at a point that is 9 cm away from 𝐡 . Find the magnitude of weight 𝑀 that should be suspended from end 𝐡 so that the rod is about to rotate, and determine the value of the pressure 𝑃 exerted on 𝐢 in that situation.

  • A 𝑀 = 3 1 . 0 5 N , 𝑃 = 1 9 7 . 0 5 N
  • B 𝑀 = 2 0 6 . 8 N , 𝑃 = 3 7 2 . 8 N
  • C 𝑀 = 1 4 6 . 4 5 N , 𝑃 = 3 1 2 . 4 5 N
  • D 𝑀 = 1 3 0 . 4 5 N , 𝑃 = 2 9 6 . 4 5 N

Q12:

𝐴 𝐡 is a uniform rod of length 76 cm and weight 69 newtons. It is suspended horizontally by two vertical strings from its two ends 𝐴 and 𝐡 . Determine the distance from 𝐴 in cm that a weight of 75 newtons should be suspended for the tension magnitude at 𝐴 to be twice its magnitude at 𝐡 .

  • A 62.32 cm from 𝐴
  • B 13.68 cm from 𝐴
  • C 83.6 cm from 𝐴
  • D 14.87 cm from 𝐴

Q13:

A uniform rod 𝐴 𝐡 having a length of 62 cm and weighing 13 N is resting horizontally by means of a support and a string. Given that the support is at the end 𝐴 and the string is 8 cm away from the end 𝐡 , determine the string’s tension 𝑇 and the support’s reaction 𝑅 , rounding your answer to two decimal places.

  • A 𝑇 = 7 . 4 6 N , 𝑅 = 2 0 . 4 6 N
  • B 𝑇 = 1 0 . 3 3 N , 𝑅 = 2 . 6 7 N
  • C 𝑇 = 6 . 5 0 N , 𝑅 = 6 . 5 0 N
  • D 𝑇 = 1 . 6 8 N , 𝑅 = 1 1 . 3 2 N
  • E 𝑇 = 7 . 4 6 N , 𝑅 = 5 . 5 4 N

Q14:

𝐴 𝐡 is a rod having a length 120 cm and weighing 12 N, which is acting at a point 15 cm away from 𝐴 . Given that the rod is resting on a support at its midpoint, determine the reaction of the support 𝑅 , and find the weight π‘Š that should be suspended from the end 𝐡 to make the rod in equilibrium in a horizontal position.

  • A π‘Š = 9 N , 𝑅 = 2 1 N
  • B π‘Š = 1 6 N , 𝑅 = 2 8 N
  • C π‘Š = 2 1 N , 𝑅 = 3 3 N
  • D π‘Š = 1 5 N , 𝑅 = 2 7 N

Q15:

Jennifer lay on a horizontal uniform wooden plank of length 2.6 m and weight 16 kg-wt that was fixed at each end on two supports 𝐴 and 𝐡 . Given that the reactions of the two supports 𝐴 and 𝐡 are 68 kg-wt and 52 kg-wt, respectively, determine the distance between the point of action of her weight and support 𝐴 .

  • A 1 1 1 0 m
  • B 9 1 0 m
  • C 3 2 m
  • D 5 3 6 3 m

Q16:

A non-uniform wooden board 𝐴 𝐡 , having a length of 16 m, is resting horizontally on two supports at 𝐢 and 𝐷 such that 𝐴 𝐢 = 3 m and 𝐡 𝐷 = 4 m . If the maximum distance that a man, whose weight is 639 N, can move on the board from 𝐴 to 𝐡 without getting the board imbalanced is 14.2 m, and the maximum distance the same man can move from 𝐡 to 𝐴 is 14.8 m, find the weight 𝑀 of the board and the distance π‘₯ between its line of action and the point 𝐴 .

  • A 𝑀 = 3 9 0 . 5 N , π‘₯ = 1 5 . 6 m
  • B 𝑀 = 1 3 0 . 2 9 N , π‘₯ = 1 2 . 9 9 m
  • C 𝑀 = 2 8 4 N , π‘₯ = 7 . 0 5 m
  • D 𝑀 = 6 9 9 . 4 N , π‘₯ = 9 . 9 9 m

Q17:

A uniform iron beam having a weight of 56 N and a length of 100 cm is resting horizontally on two supports 𝐴 and 𝐡 , where 𝐴 is at the end of the beam, and 𝐡 is 44 cm away from the other end. Determine the reactions of the supports 𝑅  and 𝑅  .

  • A 𝑅 = 3 . 3 6  N , 𝑅 = 5 2 . 6 4  N
  • B 𝑅 = 5 2 . 6 4  N , 𝑅 = 3 . 3 6  N
  • C 𝑅 = 5 0  N , 𝑅 = 6  N
  • D 𝑅 = 6  N , 𝑅 = 5 0  N

Q18:

In the figure, forces having magnitudes of 61, 43, 100, and 𝐹 newtons are acting on the light rod, and the rod is in equilibrium horizontally. Determine the length of 𝐷 𝐴 and the magnitude of 𝐹 .

  • A 𝐷 𝐴 = 3 9 . 5 c m , 𝐹 = 2 0 4 N
  • B 𝐷 𝐴 = 8 2 . 5 c m , 𝐹 = 1 0 0 N
  • C 𝐷 𝐴 = 3 9 . 5 c m , 𝐹 = 1 0 0 N
  • D 𝐷 𝐴 = 8 2 . 5 c m , 𝐹 = 2 0 4 N

Q19:

The mass of a motorbike is 237 kg, and its weight is acting at the vertical line passing through the midpoint between the two wheels, which are 128 cm apart. The mass of the biker is 73 kg, where his weight is acting vertically downwards at a distance of 96 cm away from the front wheel. Given that the forces shown in the figure are in kg-wt, find the reaction of the ground on the front wheel 𝑅  and the rear wheel 𝑅  .

  • A 𝑅 = 1 3 6 . 7 5  k g - w t , 𝑅 = 1 7 3 . 2 5  k g - w t
  • B 𝑅 = 9 5 . 7 5  k g - w t , 𝑅 = 2 1 4 . 2 5  k g - w t
  • C 𝑅 = 1 7 3 . 2 5  k g - w t , 𝑅 = 1 3 6 . 7 5  k g - w t
  • D 𝑅 = 2 1 4 . 2 5  k g - w t , 𝑅 = 9 5 . 7 5  k g - w t

Q20:

𝐴 𝐡 is a uniform rod having a length of 144 cm and a weight of 360 g-wt, and it is resting in a horizontal position on two supports 𝐢 and 𝐷 . The distance between the supports is 72 cm, and 𝐴 𝐢 = 3 0 c m . If a weight 𝑀 is hung at 𝐸 , where 𝐴 𝐸 = 3 6 c m , determine the value of 𝑀 which makes the reaction at 𝐢 twice that at 𝐷 .

Q21:

𝐴 𝐡 is a non-uniform rod having a length of 77 cm resting in a horizontal position on one support, which is 26 cm away from the end 𝐴 . It is kept in equilibrium by suspending a weight of 16 N at its end 𝐴 and a weight of 2 N at the end 𝐡 . If the distance between the support and the end 𝐴 is changed to be 23 cm, the rod will be kept in equilibrium by suspending a weight of 18 N at the end 𝐴 only. Find the magnitude of the weight of the rod π‘Š and the distance π‘₯ between its line of action and point 𝐴 . Round your answers to the nearest two decimal places.

  • A π‘Š = 3 3 . 3 3 N , π‘₯ = 1 0 . 5 8 c m
  • B π‘Š = 0 . 6 7 N , π‘₯ = 5 7 5 . 0 0 c m
  • C π‘Š = 3 3 . 3 3 N , π‘₯ = 3 5 . 4 2 c m
  • D π‘Š = 6 9 . 3 3 N , π‘₯ = 1 9 . 2 5 c m
  • E π‘Š = 3 9 2 . 6 7 N , π‘₯ = 2 4 . 0 5 c m

Q22:

A uniform rod 𝐴 𝐡 is resting horizontally at its ends on two supports, and it carries a weight of 17 N at a point that’s 96 cm away from 𝐡 . If a weight of 64 N is suspended at a point on the rod that’s π‘₯ cm away from 𝐡 , the reaction at 𝐡 is double that at 𝐴 . Given that the rod is 144 cm long and weighs 30 N, determine the value of π‘₯ and the magnitude of the reaction at 𝐴 .

  • A π‘₯ = 1 2 0 c m , 𝑅 = 3 7  N
  • B π‘₯ = 2 4 c m , 𝑅 = 7 4  N
  • C π‘₯ = 2 4 c m , 𝑅 = 3 7  N
  • D π‘₯ = 1 2 0 c m , 𝑅 = 7 4  N

Q23:

A uniform rod 𝐴 𝐡 is resting horizontally on two supports; one of them is at the end 𝐴 , and the other is at the point 𝐢 , which is 27 cm away from 𝐡 . If a weight π‘Š is suspended at 𝐡 , the reaction at the support 𝐢 becomes five times that at the support 𝐴 . Given that 𝐴 𝐡 is 108 cm and weighs 27 N, determine the weight π‘Š and the reaction 𝑅  at the support 𝐢 .

  • A 𝑅 = 3 0  N , π‘Š = 9 N
  • B 𝑅 = 6 0  N , π‘Š = 4 5 N
  • C 𝑅 = 2 7 0  N , π‘Š = 2 9 7 N
  • D 𝑅 = 3 0  N , π‘Š = 3 N
  • E 𝑅 = 5 4  N , π‘Š = 3 7 . 8 N

Q24:

A nonuniform rod 𝐴 𝐡 of length 138 cm is suspended by two vertical strings. The first string is attached to 𝐡 , and the other one is attached to a point 48 cm from 𝐴 . The tension in the string attached to 𝐡 is 1 4 of the tension of the other string. Given that the maximum weight that can be suspened from 𝐴 is 24 N, determine the weight of the rod and the distance between 𝐡 and its centre of mass.

  • A π‘₯ = 6 6 c m , π‘Š = 1 6 N
  • B π‘₯ = 7 2 c m , π‘Š = 6 4 N
  • C π‘₯ = 6 6 c m , π‘Š = 6 4 N
  • D π‘₯ = 7 2 c m , π‘Š = 1 6 N

Q25:

The given figure shows a wooden board of mass 10 kg for each metre of its length. If it rests horizontally on two supports, 𝐴 and 𝐡 , and carries a box of mass 180 kg, find the reactions 𝑅  and 𝑅  exerted by the supports 𝐴 and 𝐡 respectively.

  • A 𝑅 = 1 5 2 . 5  k g - w t , 𝑅 = 1 0 7 . 5  k g - w t
  • B 𝑅 = 7 2 . 5  k g - w t , 𝑅 = 8 2 . 5  k g - w t
  • C 𝑅 = 1 0 7 . 5  k g - w t , 𝑅 = 1 5 2 . 5  k g - w t
  • D 𝑅 = 8 2 . 5  k g - w t , 𝑅 = 7 2 . 5  k g - w t

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