Worksheet: Equilibrium of Extended Objects

In this worksheet, we will practice analyzing the net effect of the forces and torques acting on objects that have extension in one or two dimensions.

Q1:

A 3.8 m long uniform density ramp with a weight of 20 N is at rest at an angle of 40 above the horizontal, as shown in the diagram. A box is dragged up the ramp. At the point where the box is 0.65 m from the top end of the ramp, it exerts a downward force of 32 N on the ramp.

What is the magnitude of 𝑅?

What is the magnitude of 𝑅?

Q2:

A uniform-density plank is at rest at an angle above the horizontal. The plank is supported by a pivot at a point directly below its center of mass. Which of the following diagrams correctly shows the forces acting on the plank due to the pivot? The weight of the plank is shown in orange and the forces due to contact with the pivot are shown in blue.

  • A
  • B
  • C
  • D
  • E

Q3:

A perfectly rigid disk of uniform density has three forces acting on it at different points on its upper surface, as shown in the diagram. The center of mass of the disk does not move.

Find the angle 𝜃.

Find the magnitude of the force 𝐹.

Can the disk be in rotational equilibrium if no other forces act on it?

  • AThe disk cannot be in rotational equilibrium.
  • BThe disk can be in rotational equilibrium.
  • CIt is not possible to tell from the information in the question.

Q4:

A ladder stands between a wall and a floor at an angle of 52 above the horizontal, as shown in the diagram. A horizontal applied force of 44 N is acting on the ladder at a point 1.3 m along its length from the position of its center of mass.

What is the magnitude of the torque on the ladder around the point 𝑃 due to the horizontal applied force?

What is the magnitude of the torque on the ladder around the point 𝑃 due to its weight?

Q5:

A 5.0 m long, uniform-density ladder with a weight of 70 N is at rest against a wall, at an angle of 60 above the horizontal, as shown in the diagram. A friction force of 20 N acts on the end of the ladder resting on the wall.

What is the magnitude of the reaction force 𝑅?

What is the magnitude of the reaction force 𝑅?

What is the magnitude of the friction force 𝐹?

Q6:

A 2.5 m long uniform-density plank of weight 28 N is suspended by a vertical rope that applies a force F to the plank. The plank is held at rest at an angle of 23 above the horizontal by forces applied by three ropes. A rope is attached at each of the ends of the plank, and a third rope is attached at a point a distance of 𝑑 meters to the right of its center of mass, as shown in the diagram. The force at the lower end of the plank is 15 N, directed at an angle of 35 from the downward vertical, away from the plank. The force at the higher end of the plank is 18 N, directed at an angle of 𝜃 below the horizontal, away from the plank.

What is the angle 𝜃?

What is the magnitude of F?

What is the magnitude of 𝑑?

Q7:

A section of an overpass has a weight of 80 kN. The overpass section is supported by two pillars, as shown in the diagram. The pillars are 20 m and 12 m horizontally distant from the center of mass of the section. The section is tilted slightly below the horizontal at an angle of 72 from the vertical. The two pillars have different coefficients of static friction with the overpass section.

Find the magnitude of the normal reaction force F.

Find the magnitude of the normal reaction force F.

Find the ratio of the magnitude of the friction force F to the magnitude of the friction force F.

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