Forces, Work, Torque, and Oscillations

Multiple Choice and Conceptual Problems

This section covers key concepts in classical mechanics, including torque, work, conservative forces, and simple harmonic motion, as well as their applications in physical systems.

1. Torque and Rotational Equilibrium

• Torque (τ): The rotational equivalent of force, defined as the product of force and the lever arm (distance from the pivot point), and the sine of the angle between them.

• Formula:

• Rotational Equilibrium: Occurs when the sum of all torques acting on a system is zero, resulting in no angular acceleration.

• Example: A uniform meter stick balanced at its center with a mass suspended at one end. The mass required to balance the stick can be found by setting the clockwise and counterclockwise torques equal.

2. Work Done by a Variable Force

• Work (W): The energy transferred by a force acting over a distance. For a variable force, work is the area under the force vs. displacement graph.

• Formula:

• Example: For a piecewise linear force graph, calculate the area under each segment and sum them to find the total work.

3. Conservative Forces

• Definition: A force is conservative if the work it does on an object moving between two points is independent of the path taken.

• Properties:

  • Conservative forces have an associated potential energy function.

  • The total mechanical energy (kinetic + potential) of a system remains constant in the absence of non-conservative forces.

  • Examples: Gravitational force, spring force.

• Non-conservative forces: (e.g., friction) dissipate mechanical energy as heat or other forms.

4. Simple Harmonic Motion (SHM)

• Definition: Motion that repeats itself in a regular cycle, such as a mass on a spring or a pendulum for small angles.

• General Equation: , where:

  • = amplitude (maximum displacement)

  • = angular frequency ()

  • = frequency (cycles per second)

  • = period ()

• Velocity and Acceleration:

  • Velocity:

  • Acceleration:

• Example: An astronaut oscillating on a spring in space. The period and angular frequency can be determined from the graph of position vs. time.

5. Forces Acting on a Sled (Work and Energy)

• Forces on a Sled: When pulling a sled at constant speed, the following forces do negative work:

  • Gravity (if moving uphill)

  • Tension in the handle (if pulling against friction or gravity)

  • Kinetic friction (always opposes motion)

  • Normal force (perpendicular, does no work)

  • Air drag (opposes motion)

• Work: Only forces with a component along the direction of displacement do work.

6. Properties of Conservative Forces (Table)

7. Torque and Lever Arm

• Torque: Only the component of force perpendicular to the lever arm produces torque.

• Effect of Lever Arm Length: For a given force, a longer lever arm produces more torque ().

• Direction: The sign of torque depends on the direction of rotation (clockwise or counterclockwise).

8. Example Problems and Solutions

• Balancing a Meter Stick: To balance a uniform stick with a mass at one end, set the torques about the pivot equal and solve for the unknown mass.

• Work from Force Graph: Calculate the area under the force vs. position graph to find the total work done.

• Oscillating Astronaut: From the graph, determine the period (T) as the time for one complete cycle, and angular frequency () as .

9. Key Equations

• Torque:

• Work (constant force):

• Work (variable force):

• Simple Harmonic Motion:

• Angular Frequency:

10. Additional Info

• In SHM, the maximum velocity is , and the maximum acceleration is .

• At maximum displacement, velocity is zero; at equilibrium, acceleration is zero.

• Forces that act opposite to displacement (like friction and air drag) always do negative work.