Question

A point on a rotating object has an initial angular velocity ω0 and rotates with an angular acceleration α0 for a time interval from t=0 to time t=t0. The point then rotates at a constant angular speed until time t=t1. What is the angular displacement of the point from t=0 to t=t1? Express your answer in terms of ω0, α0, t0, t1, and/or any fundamental constants as appropriate.

So, the answer is apparently
ω0t0+1/2α0t20+(ω0+α0t0)t1
Why tho? Can someone show me the work? Because the way to get this answer seems to neglect the fact that the second displacement is actually w0(t1-t0)
But, should it not be

Answer 1

The angular displacement of a point on a rotating object involves calculating the displacement during the acceleration phase using the rotational kinematic equation and then adding the displacement during the constant speed phase.

Step-by-step explanation:

To calculate the angular displacement of a point on a rotating object, we need to consider two phases of motion: when the object is accelerating and when it is moving at constant speed. Let's break down the problem into these two phases.

Phase 1: Accelerating Motion

During the time from t=0 to t=t0, the point is accelerating. The angular displacement during this period (θ1) can be found using the kinematic equation for rotational motion:

θ1 = ω0t0 + ½α0t02

Phase 2: Constant Angular Velocity

After t0, the point rotates at a constant angular velocity until t=t1. The angular displacement during this period (θ2) is:

θ2 = (ω0 + α0t0)(t1 - t0)

Adding both displacements gives us the total angular displacement (θ) from t=0 to t=t1:

θ = θ1 + θ2
θ = (ω0t0 + ½α0t02) + (ω0 + α0t0)(t1 - t0)

Answer 2

The angular displacement of the point from t=0 to t=t1, considering initial angular velocity and angular acceleration, includes the sum of displacement during acceleration and at constant angular velocity, resulting in θtotal = (ω0t0 + ½α0t02) + (ω0 + α0t0)(t1 - t0).

Step-by-step explanation:

The question relates to the angular displacement of a point on a rotating object. The object has an initial angular velocity of ω0 and angular acceleration α0 during time t0, followed by a constant angular speed until time t1. The angular displacement for the first segment, when the object is accelerating, can be found using the equation θ = ω0t + ½α0t2, which gives the angular displacement during acceleration until time t0. After time t0, for the second displacement, we need to calculate the constant angular velocity, which is ω = ω0 + α0t0, because it includes the effect of the acceleration during the first segment. The angular displacement during the second part (from t0 to t1) is θ = ω(t1 - t0). Combining both segments, the total angular displacement from t=0 to t=t1 is:

θtotal = (ω0t0 + ½α0t02) + (ω0 + α0t0)(t1 - t0)

This result accounts for the angular displacement during both the acceleration and the constant angular velocity phases of the motion.