Worksheet: Rotational Equilibrium

In this worksheet, we will practice calculating the positions and masses of objects in rotational systems such that the net rotational acceleration is zero.

Q1:

A uniform seesaw is balanced on a fulcrum located 3.0 m from the left end, as shown. The smaller boy on the right has a mass of 40 kg and the bigger boy on the left has a mass of 80 kg. What is the mass of the board?

Q2:

The uniform seesaw is balanced at its center of mass, as shown. The smaller boy on the right has a mass of 40 kg. What is the mass of his friend?

Q3:

A uniform scaffold with of weight 40.0 kg and length 6.0 m is supported by two light cables, as shown in the figure. A painter weighing 80.0 kg stands 1.0 m from the left end of the scaffold, and his painting equipment is 1.5 m from the right end. The tension in the left cable is twice that in the right cable.

Find the tension in the right cable.

Find the tension in the left cable.

Find the mass of the equipment.

Q4:

The forearm shown in the diagram is positioned at an angle πœƒ with respect to the upper arm, and a ball with a mass of 5.0 kg is held in the hand. The total mass of the forearm and hand is 3.0 kg, and their center of mass is 15.0 cm from the elbow.

What is the magnitude of the force that the biceps muscle exerts on the forearm for πœƒ=60∘?

What is the magnitude of the force on the elbow joint for πœƒ=60∘?

Q5:

The coefficient of static friction between the rubber eraser of the pencil and the tabletop is πœ‡=0.80. If the force F is applied along the axis of the pencil, as shown below, what is the minimum angle at which the pencil can stand without slipping? Ignore the weight of the pencil.

  • A44∘
  • B47∘
  • C49∘
  • D53∘
  • E51∘

Q6:

To get up on the roof of a house, a person with a mass of 70.0 kg places an aluminum ladder of length 6.00 m against the house on a concrete pad, with the base of the ladder 2.00 m from the house. The ladder rests against a plastic rain gutter, which has negligible friction with the ladder. The ladder’s mass is 10.0 kg, and its center of mass is 2.00 m from the ladder’s bottom. The person is standing 3.00 m from the bottom of the ladder.

Find the magnitude of the normal reaction force on the base of the ladder.

Find the friction force on the base of the ladder.

Q7:

Two wheels 𝐴 and 𝐡 with weights 𝑀 and 2𝑀, respectively, are connected by a uniform rod with weight 𝑀2, as shown below. The wheels are free to roll on the sloped surfaces. Determine the angle that the rod forms with the horizontal when the system is in equilibrium.

Hint: There are five forces acting on the rod, which is two weights of the wheels, two normal reaction forces at points where the wheels make contacts with the wedge, and the weight of the rod.

Q8:

A uniform 4.0 m plank weighing 200.0 N rests against the corner of a wall, as shown in the figure. There is no friction at the point where the plank meets the corner.

Find the magnitude of the force that the corner exerts on the plank.

The force exerted by the corner of the wall on the plank acts at an angle from the horizontal. Find this angle.

Find the magnitude of the force that the floor exerts on the plank.

The force exerted by the floor on the plank acts at an angle from the horizontal. Find this angle.

What is the minimum coefficient of static friction between the floor and the plank that is required to prevent the plank from slipping?

Q9:

A pole is at a 90∘ bend in a power line, as shown. The pole is subjected to more shear force than poles in straight parts of the line. The tension in each line is 4.00Γ—10οŠͺ N at the angles shown. The pole is 15 m tall, has an 18 cm diameter, and can be considered to have a Young’s modulus of 0.75Γ—10 Pa.

Calculate the compression of the pole.

Find the magnitude of the bending of the pole.

Find the tension in a guy wire used to keep the pole straight. The guy wire is attached to the top of the pole at an angle of 30∘ with the vertical and is in the opposite horizontal direction to the bend in the power line.

  • A1.8Γ—10 N
  • B1.6Γ—10 N
  • C2.1Γ—10 N
  • D1.1Γ—10 N
  • E1.4Γ—10 N

Q10:

When the structure shown is supported at point 𝑃, the structure is in equilibrium. The weight of the structure is negligible.

Find the magnitude of the force 𝐹.

Find the magnitude of the force applied at 𝑃.

At what counterclockwise angle from the line directed horizontally to the right from the point 𝑃 does 𝐹 act?

Q11:

A homogeneous cube of weight 100 kN sits on a horizontal plane. Determine the minimum force F acting on the upper top edge of the cube, parallel with its surface, that is required to tip the block over.

Q12:

How is the beam configuration shown in the accompanying diagram loaded and supported?

  • AEnd-loaded, cantilevered beam
  • BCenter-loaded, cantilevered beam
  • CSimply supported, end-loaded beam
  • DCenter-loaded, simply supported beam
  • ESimply supported, cantilevered beam

Q13:

A uniform ladder has a length of 5.00 m and a weight of 400.0 N. One end of the ladder is on the floor which exerts a frictional force on the ladder’s end while the opposite end rests against a frictionless wall. The ladder’s length makes an angle of 53.0∘ with the horizontal, as shown in the diagram.

What is the reaction force from the floor on the ladder?

What is the reaction force from the wall on the ladder?

Find the minimum value of the coefficient of static friction between the ladder and the floor required for the ladder to maintain equilibrium.

Q14:

A person of mass 50.0 kg stands 1.50 m from the left-hand end of a uniform 6.00 m long scaffold of mass 70.0 kg, as shown in the diagram. Two ropes support the scaffold in equilibrium.

What is the magnitude of the tension in the left-hand rope?

What is the magnitude of the tension in the right-hand rope?

Q15:

A crane’s boom is aligned with the line 𝐴𝐡, as shown in the diagram. The boom is 15.0 m in length and has a mass of 1.10Γ—10 kg. The center of mass of the boom is at its geometric center. The crane lifts a load that has a mass of 2.00Γ—10 kg.

What is the magnitude of the tension 𝑇 in the cable that supports the boom?

What is the magnitude of the force exerted on the boom by the axle at point 𝐴?

Q16:

A 1.0 m wide uniform door of weight 400.0 N is suspended on a pair of hinges at the points 𝐴 and 𝐡, separated by 2.0 m, as shown in the diagram. The door is opened at an angle of 45∘ from its closed position and the hinges support the door’s weight, holding the door vertically. Assume that each hinge exerts an equal magnitude force to the other hinge.

What is the force vector of the force applied by the door to the upper hinge?

  • A(100βˆ’100)ijN
  • B(βˆ’100βˆ’200)ijN
  • C(100βˆ’200)ijN
  • D(100+100)ijN
  • E(100+200)ijN

What is the force vector of the force applied by the door to the lower hinge?

  • A(100+100)ijN
  • B(100βˆ’200)ijN
  • C(βˆ’100βˆ’200)ijN
  • D(100+200)ijN
  • E(100βˆ’100)ijN

Q17:

A uniform horizontal strut weighs 500.0 N. One end of the strut is attached to a hinged support at the wall. The opposite end of the strut is a point from which a sign is held that weighs 300.0 N. The strut is also supported by a cable that runs between the end of the strut where the sign is attached and the wall, as shown in the diagram.

Find the magnitude of tension in the cable.

Find the magnitude of the force exerted at the hinge of the strut.

Q18:

A uniform meter stick of mass 150.0 g is balanced on a fulcrum that is located 30.0 cm from the stick’s right-hand end. A weight of mass π‘š=50.0 g is suspended from the stick’s left-hand end, and a weight of mass π‘š=75.0 g is suspended 30.0 cm from the stick’s left-hand end, as shown in the diagram. A weight of an unknown mass π‘šοŠ© that is suspended from the stick’s right-hand end maintains the equilibrium of the stick-and-weights system.

Find π‘šοŠ©.

What is the magnitude of the force exerted by the tip of the fulcrum on the meter stick?

Q19:

A uniform boom weighs 4,000 N. The boom is supported by a horizontal guy wire from which is suspended a 3,000 N weight. The guy wire holds the boom at an angle of 50∘ counterclockwise above the horizontal. The boom is also supported by a hinged support at a point 𝐴 as shown in the diagram.

What is the magnitude of the force exerted by the guy wire on the boom?

What is the magnitude of the force exerted on the boom by the support at point 𝐴?

At what direction above the horizontal does the force from the support at point 𝐴 act?

  • A60∘ counterclockwise
  • B40∘ counterclockwise
  • C80∘ counterclockwise
  • D80∘ counterclockwise
  • E50∘ counterclockwise

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