FIGURE EX4.24 shows the angular-position-versus-time graph for a particle moving in a circle. What is the particle's angular velocity at (b) t = 4s

Verified step by step guidance

1. Identify the time interval in which t = 4s falls. From the graph, it is within the interval from 0s to 5s.
2. Determine the angular position (θ) at the beginning and end of this interval. At t = 0s, θ = 0 rad, and at t = 5s, θ = 2π rad.
3. Calculate the change in angular position (Δθ) over this interval: Δθ = θ_final - θ_initial = 2π rad - 0 rad.
4. Determine the time interval (Δt) over which this change occurs: Δt = 5s - 0s = 5s.
5. Calculate the angular velocity (ω) using the formula ω = Δθ / Δt. Substitute the values: ω = (2π rad) / (5s).

Key Concepts

Angular Position

Angular position refers to the angle at which an object is located in a circular path, measured in radians. In the context of the graph, it shows how the angle changes over time as the particle moves in a circle. The vertical axis represents angular position, indicating the particle's orientation at any given moment.

Angular Velocity

Angular velocity is the rate of change of angular position with respect to time, typically expressed in radians per second. It can be calculated by determining the slope of the angular position versus time graph. At t = 4s, the angular velocity can be found by analyzing the graph's slope during that interval, which indicates how quickly the particle is rotating.

Graph Interpretation

Interpreting graphs is crucial in physics for understanding relationships between variables. In this case, the angular position versus time graph allows us to visualize how the particle's position changes over time. By examining the graph's segments, we can identify periods of constant angular velocity and periods of acceleration or deceleration, which are essential for solving the problem.