A solid body rotates about a fixed axis according to the law θ=6 t-2 t^s. Ilere, θis in radian a

step 1 Hello everyone, we are going to understand this question. Here, given in the question, the body

A solid body rotates about a fixed axis according to the law...

A solid body rotates about a fixed axis according to the law $\theta=6 t-2 t^{s}$. Ilere, $\theta$ is in radian and $t$ is seconds. Find : (a) the mean values of the angular velocity and angular acceleration averaged over the time interval between $t=0$ and the complete stop, (b) the angular acceleration at the moment when the body stops.

A solid body rotates about a fixed axis according to the law $\theta=6 t-2 t^{s}$. Ilere, $\theta$ is in radian and $t$ is seconds. Find :
(a) the mean values of the angular velocity and angular acceleration averaged over the time interval between $t=0$ and the complete stop,
(b) the angular acceleration at the moment when the body stops.

Key Concepts

Angular Velocity

Angular velocity represents the rate at which the angular displacement changes with time. It is the first derivative of the angular displacement with respect to time and provides a measure, typically in radians per second, of how fast the body is rotating at any instant. In problems involving rotational motion, determining when the angular velocity reaches zero can identify points of rest or changes in motion.

Stopping Condition

The stopping condition in rotational motion is the instant when the angular velocity becomes zero, indicating that the rotating body has come to a complete halt. Analyzing this moment is crucial for determining subsequent values of angular acceleration and for computing mean values over the interval of motion leading up to the stop.

Average (Mean) Quantities in Rotational Motion

The average or mean values of angular velocity or acceleration over a time interval are computed by dividing the net change in the respective quantity by the elapsed time. For instance, the average angular velocity is calculated by taking the total angle rotated over the time interval and dividing it by the duration of that interval. This concept helps in summarizing the overall performance of the motion during a specified period.

Angular Displacement

Angular displacement is a measure of the angle through which a body rotates about a fixed axis. It is described as a function of time and represents the total orientation change of the rotating object. In rotational kinematics, the displacement function is the foundation from which velocity and acceleration are derived through differentiation.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity with time and is obtained as the second derivative of the angular displacement function. It quantifies how quickly the angular velocity increases or decreases and is a key parameter when analyzing changes in rotational motion, especially at instants where the motion transitions, such as at a complete stop.

Transcript

00:01 Hello everyone, we are going to understand this question.

00:05 Here, given in the question, the body is rotating with certain angular acceleration and angular speed.

00:15 And here it is given theta is equal to 6t minus 2 into t cube.

00:24 Now, in the part a, we have to calculate main value of angular velocity and angular acceleration averst over the time interval between t is equal to 0 and when body completely stops.

00:39 Now angular velocity omega is equal to d theta upon dt.

00:44 So after differentiation we will get 6 minus 6 into t square.

00:51 When body at t is equal to 0, omega is equal to 6 radian per second and when body was completely stopped at that time omega is equal to 0 then t is equal to 6 upon 6 square root so we will get 1 second so at 1 second body will completely stop now calculating the average angular speed and average angular acceleration so average angular speed that is represented by omega avg so integration limit is from 0 to 1 second omega into d t upon 0 to 1 second this is t is equal to 1 second and here d t now omega is equal to substituting the value omega is 6 into t minus 2 into t square and integration limit is from 0 to t into d t upon 0 to 1 and d t now doing further integration we will get 6 t square minus 2 t cube upon 3 and here integration limit is from 0 to 1 upon t and integration limit is from 0 to 1.

02:46 Now substituting the value in this.

02:51 So we will get 6 minus 2 upon 3, 6 minus sorry here we did a calculation mistake, let's correct it.

03:10 This is 6 minus 6 t square, 6 minus 6 t squared.

03:18 That is the value of omega.

03:20 Here we have already calculated 6 minus 6 minus 6 minus 6 into t square...

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