Initially a wheel is turning at 65 rad / s. 'Ihereafter, angular acceleration is given by α=(-10

step 1 So let's solve this question. So in this question we have alpha equals to minus 10 minus 5 t tha

Initially a wheel is turning at 65 rad / s. 'Ihereafter,...

Initially a wheel is turning at $65 \mathrm{rad} / \mathrm{s}$. 'Ihereafter, angular acceleration is given by $\alpha=(-10-5 t) \mathrm{rad} / \mathrm{s}^{2}$ where $t$ is the elapsed tine. (a) Find its angular speed at $t=3.0 \mathrm{~s}$. (b) With what angular displacement does it turn in these $3 \mathrm{~s}$ ?

Initially a wheel is turning at $65 \mathrm{rad} / \mathrm{s}$. 'Ihereafter, angular acceleration is given by $\alpha=(-10-5 t) \mathrm{rad} / \mathrm{s}^{2}$ where $t$ is the elapsed tine.
(a) Find its angular speed at $t=3.0 \mathrm{~s}$.
(b) With what angular displacement does it turn in these $3 \mathrm{~s}$ ?

Key Concepts

Integration in Kinematics

Integration in kinematics is used to derive one kinematic variable from another when the rate of change is known. Specifically, when dealing with rotational motion, integrating the angular acceleration yields the angular velocity, and integrating the angular velocity yields the angular displacement. This technique is essential, especially in cases where the acceleration is not constant but is given as a function of time.

Angular Displacement

Angular displacement represents the total angle through which an object rotates over a period of time, measured in radians. It is found by integrating the angular velocity function over time. In solving such problems, the initial angular position may be taken as zero, and the displacement is calculated over the specific time interval using definite integration.

Angular Velocity

Angular velocity refers to the rate of change of angular displacement and is measured in radians per second (rad/s). It is obtained by integrating the angular acceleration over time. In problems with non-constant angular acceleration, finding the angular velocity at a given moment involves integrating the acceleration function while considering the initial angular velocity as the integration constant.

Angular Acceleration

Angular acceleration is the rate at which the angular velocity of a rotating object changes over time. In many problems, including those with time-dependent acceleration, knowing the form of angular acceleration is crucial because it directly influences the dynamics of the rotation. It is typically denoted by the symbol ? and expressed in radians per second squared (rad/s²).

Please give Ace some feedback