Matter & Interactions 2nd ed. Practice Problems
Aaron Titus | High Point University
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(N)=# of solutions
2k80001     Torque on a beam     2k80001
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A 1.5-m beam has a 1-kg object hanging from its end. The beam is held in equilibrium at an angle of with respect to the horizontal, as shown below. To hold the beam in equilibrium, you grasp a handlebar of radius 2 cm at the bottom of the beam and exert a tangential force on the handlebar.

Figure: A beam in equilibrium.

  1. If the mass of the beam is negligible, what is the torque by the object on the beam, about the axis of rotation O?
  2. What magnitude force must you apply tangentially at the handlebar in order to keep the beam in equilibrium?




(a) The torque on the beam by the object is given by where is the position where the force of the object acts on the beam. Since the object is in equilibrium, then the force of the beam on the object balances the gravitational force on the object. As a result, the object exerts a force on the beam that is equal to its weight, . The position vector is .

The torque of the object on the beam is

This can also be determined by calculating the magnitude of the torque and using the right-hand rule to get the direction of the torque. Sketch the vectors and tail to tail.

Figure: Position and force vectors drawn tail to tail.

The angle between the vectors is . The magnitude of the torque of the object on the beam is

Use your right hand to determine the direction. Point your fingers in the direction of and rotate them toward as shown below. Your thumb points into the page, so the torque is in the –z direction, with a magnitude 13.8 m. As a result, the torque on the beam is .

Figure: Right-hand rule.

(b) If you are holding the beam in equilibrium, then your hands must exert a torque on the beam that balances the torque due to the object. In other words, since the beam is in equilibrium, .

Thus, your hands exert a torque on the beam that is equal in magnitude to the torque of the object on the beam, but opposite in direction. For a force applied tangent to a circle or radius R, . Then, the force applied your your hands is

Note that the force of the hands on the handlebar is MUCH larger than the force of the object on the beam. The reason is that its lever arm (0.02 m) is much smaller than the lever arm of the object ( .