Question

Cosummarize What can you say about a system that (a) has no linear acceleration and (b) has no angular acceleration?

Short answer

The system is stable and in equilibrium, with no change in velocity or rotational speed.

Step-by-step solution

Analyze Linear Acceleration

If a system has no linear acceleration, it means that the net external force acting on the system is zero. According to Newton's first law of motion, this implies the system is either at rest or moving with a constant velocity. Mathematically, this can be expressed as \( F_{net} = 0 \).

Analyze Angular Acceleration

If a system has no angular acceleration, it suggests that there is no net external torque acting on the system. According to rotational dynamics, this means the system is either not rotating or rotating at a constant angular velocity. This can be represented as \( \tau_{net} = 0 \).

Conclude the State of the System

In the absence of both linear and angular accelerations, the system is in a state of equilibrium. For translation, it maintains constant velocity (potentially zero). For rotation, it maintains constant angular velocity (possibly zero). Therefore, the system is stable and experiencing no change in motion.

Key concepts

Linear Acceleration

Linear acceleration refers to the change in velocity of an object moving along a straight path. When a system experiences no linear acceleration, it indicates stability. This is because the net external force acting on the system is zero.

• According to Newton's first law of motion, if no acceleration occurs, the object is either at rest or moving with a constant velocity.
• This is reflected mathematically as \( F_{net} = 0 \).

Applying this helps understand conditions where forces are balanced, leading to equilibrium in linear motion.

Angular Acceleration

Angular acceleration describes how quickly an object's rotational speed is changing. No angular acceleration occurs when there is no net external torque. This indicates the rotational forces are balanced:

• The system could be stationary or rotating at a constant speed.
• This can be expressed with the equation \( \tau_{net} = 0 \).

By understanding the absence of angular acceleration, we recognize rotational equilibrium, aiding in analyzing rotational dynamics.

Newton's First Law of Motion

Newton's first law of motion, often called the law of inertia, explains an object's tendency to resist changes in its state of motion. This law underpins both concepts of linear and angular equilibrium:

• An object will remain at rest or in uniform motion in a straight line unless acted upon by a net external force.
• Similarly, a rotating object will maintain its rotational state unless a net external torque is present.

Newton's first law helps predict motion stability in both linear and rotational movement, forming a base for equilibrium analysis.

Net Force

The net force is the sum of all external forces acting on a system. For a state of equilibrium in terms of linear motion, the net force must be zero.

• This scenario indicates no linear acceleration.
• The object either remains stationary or continues to move at a constant speed along a straight path.

Understanding net force is crucial in determining whether a system will change velocity or maintain its current state of uniform motion.

Net Torque

Net torque refers to the overall effect of rotational forces applied to an object. In rotational equilibrium, the net torque is zero.

• This implies no changes in the rotational speed of the object.
• This principle allows us to understand situations where an object's rotational state remains unaltered.

Recognizing when net torque is zero helps identify conditions for rotational stability and the absence of angular acceleration.