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Worksheet: Equilibrium of a Rigid Body

In this worksheet, we will practice solving problems on equilibrium of rigid bodies.

Q1:

𝐴 𝐡 is a uniform ladder with weight 177 N. End 𝐴 rests on rough horizontal ground and end 𝐡 rests against a smooth vertical wall. The ladder is inclined to the horizontal at an angle of 6 0 ∘ , and the coefficient of friction between the ground and the ladder is √ 3 4 . Find the maximum weight that can be suspended from 𝐡 without the ladder slipping.

Q2:

A uniform rod 𝐴 𝐡 of weight 10 N and length 12.5 m is resting with its end 𝐴 on a rough horizontal plane and point 𝐢 (between 𝐴 and 𝐡 ) resting against a smooth horizontal nail, which is 5.7 m above the horizontal plane. If the rod is about to slide when it is inclined to the horizontal at an angle whose tangent is 3 4 , determine the coefficient of friction between the rod and the horizontal plane.

  • A 5 6
  • B 1 0 1 1
  • C 1 0 1 9
  • D 6 1 1

Q3:

In the figure, a uniform rod of length 104 cm and weight 8 N is attached to a hinge against a vertical wall, while its free end is attached by a light rope to a point on the wall directly above the hinge. Another body weighing 6 N is suspended from the rod as shown in the figure. If the rod is in a horizontal static equilibrium, find the magnitude of the reaction of the wall to the hinge rounded to one decimal place and its direction from the horizontal rounded to the nearest minute.

  • A 𝑅 = 7 . 1 N , πœƒ = 3 9 5 9 β€² ∘
  • B 𝑅 = 1 4 . 2 N , πœƒ = 5 0 1 β€² ∘
  • C 𝑅 = 4 . 1 N , πœƒ = 3 9 5 9 β€² ∘
  • D 𝑅 = 7 . 1 N , πœƒ = 5 0 1 β€² ∘

Q4:

A ladder 𝐴 𝐡 weighing 34 kg-wt and having a length of 14 m is resting in a vertical plane with its end 𝐡 on a smooth floor and its end 𝐴 against a smooth vertical wall. The end 𝐡 , which is 3.3 m away from the wall, is attached by a string to a point on the floor directly below 𝐴 . Given that the weight of the ladder is acting on the ladder itself at a point 5.6 m away from 𝐡 , find the tension in the string when a man of weight 74 kg-wt stands on the mid-point of the ladder.

  • A 40.3 kg-wt
  • B 21.25 kg-wt
  • C 22.57 kg-wt
  • D 12.27 kg-wt

Q5:

A uniform ladder 𝐴 𝐡 having a length 𝐿 and weighing 40 kg-wt is resting with one of its ends on a smooth floor and the other against a smooth vertical wall. The ladder makes an angle of 4 5 ∘ with the horizontal, and its lower end 𝐴 is attached to a string that is fixed to a point at the junction of the wall and floor. Given that the maximum tension the string can withstand is 60 kg-wt, find how far up the ladder a man of weight 140 kg-wt can go before the string breaks.

  • A 1 4 𝐿
  • B 1 7 𝐿
  • C 1 2 𝐿
  • D 2 7 𝐿

Q6:

A uniform ladder of weight 72 N is resting with its upper end against a smooth vertical wall and with its lower end against rough horizontal ground, where the coefficient of friction between the ladder and the ground is √ 3 5 . If a force of magnitude 12 N acts on the lower end of the ladder trying to move it away from the wall in a direction upwards of the horizontal, where the force makes an angle of 3 0 ∘ with the horizontal, the ladder is about to slide. Determine the tangent of the angle that the ladder makes with the horizontal ground.

  • A 5 √ 3 4
  • B 1 0 √ 3 3
  • C √ 3 5
  • D 5 √ 3 3

Q7:

A uniform ladder weighing 25 kg-wt is resting with its upper end against a smooth vertical wall and its lower end on rough horizontal ground. The ladder lies in a plane perpendicular to the wall, and is inclined to the horizontal at an angle of 4 5 ∘ . A man, whose weight is 76 kg-wt, climbs the ladder until reaching the point 1 4 of the way up. The ladder is now on the point of sliding. If the man wanted to climb all the way to the top of the ladder, find the minimum horizontal force required to act on the lower end of the ladder in order to stop it from sliding.

  • A 18.75 kg-wt
  • B 19 kg-wt
  • C 95 kg-wt
  • D 57 kg-wt

Q8:

A uniform ladder is resting in a vertical plane with its upper end against a smooth vertical wall and its lower one on a rough horizontal floor, where the coefficient of friction between the ladder and the floor is 2 3 . The ladder is inclined to the horizontal at an angle measuring 4 8 ∘ . Given that the ladder weighs 295 N and and has a length 𝐿 , find, in terms of 𝐿 , the maximum distance a man weighing 610 N can climb up the ladder without it slipping, rounding your answer to two decimal places.

  • A 0 . 6 1 𝐿
  • B 0 . 6 5 𝐿
  • C 0 . 1 2 𝐿
  • D 0 . 8 6 𝐿

Q9:

𝐴 𝐡 is a uniform ladder weighing 51 N whose end 𝐴 is resting on rough horizontal ground, while its end 𝐡 is against a rough vertical wall so that the ladder is inclined to the horizontal at an angle of 4 5 ∘ . The coefficients of friction at 𝐴 and 𝐡 are 7 9 and 1 4 respectively. If the end 𝐴 of the ladder is pulled by a horizontal force 𝐹 to bring the ladder to the brink of moving away from the wall, find the magnitude of 𝐹 .

Q10:

A uniform ladder with length 𝐿 and weight 35 kg-wt is resting with its upper end against a rough vertical wall and lower end on a rough horizontal floor. The floor and wall both have the same coefficient of friction of 1 2 with the ladder. If the ladder is inclined to the horizontal at an angle whose tangent is 3 4 , find the maximum distance (in terms of 𝐿 ) that a man weighing 78 kg-wt can ascend the ladder without it slipping.

  • A 1 4 𝐿
  • B 2 3 𝐿
  • C 3 4 𝐿
  • D 1 2 𝐿

Q11:

A uniform rod 𝐴 𝐡 weighing 111 N is resting in a vertical plane with its upper end 𝐴 against a smooth vertical wall and its lower end 𝐡 on a rough horizontal floor. If the rod is resting in limiting equilibrium when inclined by an angle of 3 0 ∘ to the horizontal, find the coefficient of friction πœ‡ between the rod and the floor and the reaction of the wall 𝑅 π‘Ž at its upper end 𝐴 rounded to two decimal places.

  • A πœ‡ = 0 . 5 8 , 𝑅 = 6 4 . 0 9 π‘Ž N
  • B πœ‡ = 0 . 2 9 , 𝑅 = 3 2 . 0 4 π‘Ž N
  • C πœ‡ = 0 . 8 7 , 𝑅 = 1 2 8 . 1 7 π‘Ž N
  • D πœ‡ = 0 . 8 7 , 𝑅 = 9 6 . 1 3 π‘Ž N
  • E πœ‡ = 0 . 2 9 , 𝑅 = 3 8 4 . 5 2 π‘Ž N

Q12:

𝐴 𝐡 is a uniform rod of length 32 cm and weight 255 kg-wt, where end 𝐴 is attached to a hinge fixed to a vertical wall, and end 𝐡 is held by a light string whose other end 𝐢 is fixed to the wall 24 cm vertically above 𝐴 . Given that the rod is held horizontally in equilibrium, find the magnitude of the tension in the string, 𝑇 , the reaction of the hinge, 𝑅 , and the measure of the angle πœƒ made between the reaction of the hinge and the rod, stating your answer to the nearest minute.

  • A 𝑇 = 4 2 5 k g - w t , 𝑅 = 2 1 2 . 5 k g - w t , πœƒ = 5 3 8 β€² ∘
  • B 𝑇 = 2 1 2 . 5 k g - w t , 𝑅 = 4 2 5 k g - w t , πœƒ = 3 6 5 2 β€² ∘
  • C 𝑇 = 4 2 5 k g - w t , 𝑅 = 4 2 5 k g - w t , πœƒ = 5 3 8 β€² ∘
  • D 𝑇 = 2 1 2 . 5 k g - w t , 𝑅 = 2 1 2 . 5 k g - w t , πœƒ = 3 6 5 2 β€² ∘

Q13:

A uniform rod 𝐴 𝐡 with a weight of 134 N is rotating freely about a hinge at 𝐡 , and its end 𝐴 is resting against a smooth vertical wall. The rod is inclined to the vertical at an angle of 4 5 ∘ , and it sits in equilibrium in a vertical plane perpendicular to the wall. Find the magnitude of the reaction of the rod 𝑅 𝐴 at the point 𝐴 , the magnitude of the reaction 𝑅 𝐡 at the hinge 𝐡 , and the tangent of the angle πœƒ that the reaction 𝑅 𝐡 makes with the horizontal.

  • A 𝑅 = 1 3 4 𝐴 N , 𝑅 = 1 3 4 √ 2 𝐡 N , t a n πœƒ = 1
  • B 𝑅 = 6 7 𝐴 N , 𝑅 = 6 7 √ 5 𝐡 N , t a n πœƒ = 1 2
  • C 𝑅 = 6 7 𝐴 N , 𝑅 = 2 0 1 𝐡 N , t a n πœƒ = 2
  • D 𝑅 = 6 7 𝐴 N , 𝑅 = 6 7 √ 5 𝐡 N , t a n πœƒ = 2
  • E 𝑅 = 1 3 4 𝐴 N , 𝑅 = 2 6 8 𝐡 N , t a n πœƒ = 1

Q14:

𝐴 𝐡 is a uniform rod of weight 88 kg-wt. It is attached to a vertical wall at end 𝐴 by means of a hinge. It is held in equilibrium by a light string connecting 𝐡 to point 𝐢 which is located on the wall vertically above point 𝐴 . Given that 𝐴 𝐡 = 𝐴 𝐢 = 𝐡 𝐢 , find the tension 𝑇 in the string and the reaction 𝑅 of the hinge on the rod.

  • A 𝑇 = 8 8 k g - w t , 𝑅 = 4 4 √ 3 k g - w t
  • B 𝑇 = 4 4 √ 3 k g - w t , 𝑅 = 4 4 k g - w t
  • C 𝑇 = 4 4 k g - w t , 𝑅 = 4 4 √ 2 k g - w t
  • D 𝑇 = 4 4 k g - w t , 𝑅 = 4 4 √ 3 k g - w t

Q15:

𝐴 𝐡 is a nonuniform rod of length 24 cm and weight 11 kg-wt acting at point 𝐷 on the rod, where 𝐴 𝐷 = 2 1 c m . The rod is attached to a vertical wall by means of a hinge at 𝐴 . The other end of the rod, 𝐡 , is tied to a light string whose other end is fixed to the wall at point 𝐢 lying 70 cm vertically above 𝐴 . If the rod is in equilibrium when it is perpendicular to the wall, find the tension 𝑇 in the string and the reaction 𝑅 of the hinge.

  • A 𝑇 = 1 3 . 0 8 k g - w t , 𝑅 = 1 0 . 1 8 k g - w t
  • B 𝑇 = 1 . 4 5 k g - w t , 𝑅 = 7 . 1 5 k g - w t
  • C 𝑇 = 2 3 . 2 6 k g - w t , 𝑅 = 1 2 . 9 3 k g - w t
  • D 𝑇 = 1 0 . 1 7 5 k g - w t , 𝑅 = 3 . 5 7 5 k g - w t

Q16:

𝐴 𝐡 is a uniform rod of length 78 cm and weight 41 N that can rotate without resistance in a vertical plane about a hinge attached to a vertical wall at 𝐴 . The rod passes through a smooth ring tied by a thin string having a length of 9 cm. The other end of the string is fixed at point 𝐢 that lies vertically above 𝐴 at a distance of 15 cm. The Rod Rests in equilibrium and the string is perpendicular to the rod. Find the tension 𝑇 in the string and the direction of the reaction of the hinge represented by an angle πœƒ with the horizontal to the nearest minute.

  • A 𝑇 = 6 3 . 9 6 N , πœƒ = 2 5 6 β€² ∘
  • B 𝑇 = 7 9 . 9 5 N , πœƒ = βˆ’ 8 3 4 7 β€² ∘
  • C 𝑇 = 1 3 3 . 2 5 N , πœƒ = βˆ’ 2 0 4 β€² ∘
  • D 𝑇 = 7 9 . 9 5 N , πœƒ = βˆ’ 6 1 3 β€² ∘
  • E 𝑇 = 7 9 . 9 5 N , πœƒ = βˆ’ 2 5 3 5 β€² ∘

Q17:

A uniform rod weighing 8 N is resting on two smooth inclined planes. The first plane is inclined at 5 0 ∘ to the horizontal, while the second is inclined at 4 0 ∘ to the horizontal. Determine the angle πœƒ the rod makes with the horizontal when it is in a state of equilibrium and the forces the rod exerts on each plane 𝑃 1 and 𝑃 2 , stating your answer to the nearest two decimal places.

  • A 𝑃 = 6 . 1 3 1 N , 𝑃 = 5 . 1 4 2 N , πœƒ = 8 0 . 0 0 ∘
  • B 𝑃 = 6 . 1 3 1 N , 𝑃 = 5 . 1 4 2 N , πœƒ = 1 0 . 0 0 ∘
  • C 𝑃 = 5 . 1 4 1 N , 𝑃 = 6 . 1 3 2 N , πœƒ = 8 0 . 0 0 ∘
  • D 𝑃 = 5 . 1 4 1 N , 𝑃 = 6 . 1 3 2 N , πœƒ = 1 0 . 0 0 ∘

Q18:

A uniform rod 𝐴 𝐡 of weight 73 N is resting with its end 𝐴 on a smooth horizontal plane and its other end 𝐡 against a smooth inclined plane. The inclined plane makes an angle of 6 0 ∘ with the horizontal one. A horizontal string prevents the rod from sliding, where one of its ends is attached to the end 𝐴 of the rod, and the other is fixed to the junction of the two planes, such that the rod and the string are in the same vertical plane, which is perpendicular to the line of junction of the two planes. If the rod is about to slide when it is inclined to the horizontal plane at an angle of 3 0 ∘ , determine the reaction of the inclined plane on the rod 𝑅 and the tension in the string 𝑇 .

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

Q19:

A rod of length 40 cm rotates without resistance in a vertical plane about a hinge that is fixed at its end. A couple of moment 4 1 √ 2 kg-wtβ‹…cm, whose direction is perpendicular to the vertical plane in which the rod rotates, acts on the rod. Given that the weight of the rod, which is 4.1 kg-wt, acts at its midpoint, identify the magnitude of the reaction of the hinge 𝑅 and the angle πœƒ of inclination of the rod to the vertical in equilibrium position.

  • A 𝑅 = 4 1 k g - w t , πœƒ = 7 5 ∘
  • B 𝑅 = 4 . 1 k g - w t , πœƒ = 7 5 ∘
  • C 𝑅 = 4 1 k g - w t , πœƒ = 1 3 5 ∘
  • D 𝑅 = 4 . 1 k g - w t , πœƒ = 4 5 ∘

Q20:

A uniform rod 𝐴 𝐡 weighing 8 N is freely moving in a vertical plane about a fixed hinge at 𝐴 . A couple with a moment of 106 Nβ‹…cm is acting on it in its plane. If the rod rests in equilibrium when it is inclined above the horizontal at an angle of 3 0 ∘ , find the length of the rod.

  • A 53 cm
  • B 5 3 √ 3 6 cm
  • C 5 3 2 cm
  • D 5 3 √ 3 3 cm

Q21:

A uniform ladder having a weight of 140 N and a length of 7 m rests with one end 𝐴 on smooth horizontal ground and its other end 𝐡 on a smooth vertical wall. The ladder is kept in equilibrium by attaching its end 𝐴 to a string fixed to the junction of the wall and the ground vertically below 𝐡 . The ladder is inclined to the horizontal at an angle of 4 5 ∘ . If a man whose mass is 85 kg climbs the ladder to the point 4.9 m away from end 𝐴 , find the tension in the string rounded to 2 decimal place. Use the value 𝑔 = 9 . 8 / m s 2 for gravitational acceleration.

Q22:

𝐴 𝐡 is a uniform wooden beam having a length of 62 m and a weight of 50 kg-wt. It is resting in a horizontal position on two supports 𝐢 and 𝐷 , where 𝐴 𝐢 = 1 1 m and 𝐡 𝐷 = 2 5 m . Starting from end 𝐴 , a man weighing 75 kg-wt walks along the beam, towards end 𝐡 . Find how far he can move without the beam tipping.

Q23:

A uniform rod 𝐴 𝐡 , weighing 55 kg-wt and having a length of 160 cm, is freely hinged at 𝐴 to a vertical wall, and a weight which equals that of the rod is hung from its other end 𝐡 . The rod is kept in a horizontal position by a light inextensible rope tied to a point that is 128 cm away from 𝐴 , while the other end of the rope is fixed to the wall at a point above 𝐴 . Given that the rope is inclined to the horizontal by 6 0 ∘ , determine the tension 𝑇 in the rope and the reaction 𝑅 of the hinge, rounding your answers to two decimal places.

  • A 𝑇 = 5 2 . 9 2 N , 𝑅 = 3 7 . 4 2 N
  • B 𝑇 = 2 0 6 . 2 5 N , 𝑅 = 2 8 6 . 0 4 N
  • C 𝑇 = 4 7 6 . 3 1 N , 𝑅 = 3 4 9 . 0 5 N
  • D 𝑇 = 1 1 9 . 0 8 N , 𝑅 = 5 9 . 9 3 N
  • E 𝑇 = 7 6 . 2 1 N , 𝑅 = 1 5 7 . 9 5 N

Q24:

A uniform beam 𝐴 𝐡 of weight 106 N is resting with its end 𝐴 on a rough horizontal ground and with its end 𝐡 against a rough vertical wall, where the coefficient of friction between the beam and the wall is four times that between the beam and the ground. If the beam is about to move when it is inclined to the wall at an angle whose tangent is 1 4 4 5 , determine the reaction of the wall rounded to two decimal places.

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