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5.5 Rotational Equilibrium and Newton’s First Law in Rotational Form

Syllabus
2024
Topic
5.5
Level
—

Learning objectives

5.5A—Describe the conditions under which a system’s angular velocity remains constantDescribe the conditions under which a system’s angular velocity remains constant.• A system may exhibit rotational equilibrium (constant angular velocity) without being in translational equilibrium, and vice versa.- i. Free-body and force diagrams describe the nature of the forces and torques exerted on an object or rigid system.- ii. Rotational equilibrium is a configuration of torques such that the net torque exerted on the system is zero. Relevant equation:- iii. The rotational analog of Newton’s first law is that a system will have a constant angular velocity only if the net torque exerted on the system is zero.• A rotational corollary to Newton’s second law states that if the torques exerted on a rigid system are not balanced, the system’s angular velocity must be changing. BOUNDARY STATEMENT AP Physics 1 does not expect students to simultaneously analyze rotation in multiple planes. | TOPIC 5.6 Newton’s Second Law in Rotational Form

Recognize rotational equilibrium

Connect net torque to angular velocity

Newton's first law in rotational form: a system's angular velocity remains constant only when the net torque about the chosen axis is zero. Constant angular velocity can mean no rotation or steady nonzero rotation.

∑iτi=0\sum_i \tau_i=0

1

Check torque balance

Use a force/torque diagram to check rotational equilibrium:

  1. Choose one axis and one rotation plane.
  2. Assign opposite signs to opposite rotational senses.
  3. Calculate each torque about that axis.
  4. Add the signed torques. If the sum is zero, angular velocity is constant; otherwise it changes.

Separate rotational and translational equilibrium

Condition What remains constant? Required net quantity
Rotational equilibrium Angular velocity ∑τ=0\sum\tau=0
Translational equilibrium Linear velocity ∑F⃗=0\sum\vec F=0
Both Both velocities Both sums are zero

Balance two nonzero torques

Balanced torques: take counterclockwise as positive. A perpendicular 25 N25\,\text{N} force at 0.40 m0.40\,\text{m} gives −10 N m-10\,\text{N}\,\text{m}, while a perpendicular 10 N10\,\text{N} force at 1.0 m1.0\,\text{m} gives +10 N m+10\,\text{N}\,\text{m}.

∑τ=+10−10=0\sum\tau=+10-10=0, so the angular velocity stays constant even though both torques are nonzero.

Do not confuse equilibrium with rest

Equilibrium does not require rest: zero net torque means no angular acceleration, not necessarily zero angular velocity. A system can be in rotational equilibrium without translational equilibrium, or vice versa. AP Physics 1 does not require simultaneous rotation analysis in multiple planes.