Question
The angular acceleration of a wheel is $\alpha=6.0 t^{4}-4.0 t^{2}$, with $\alpha$ in radians per second-squared and $t$ in seconds. At time $t=0$, the wheel has an angular velocity of $+2.5 \mathrm{rad} / \mathrm{s}$ and an angular position of $+1.5 \mathrm{rad}$. Write expressions for (a) the angular velocity ( $\mathrm{rad} / \mathrm{s}$ ) and (b) the angular position (rad) as functions of time (s).
Step 1
The angular velocity is the integral of the angular acceleration with respect to time. So, we integrate $\alpha=6.0 t^{4}-4.0 t^{2}$ with respect to time $t$. \[ \omega = \int \alpha \, dt = \int (6.0 t^{4}-4.0 t^{2}) \, dt \] Show more…
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The angular acceleration of a wheel is $\alpha=6.0 t^{4}-4.0 t^{2},$ with $\alpha$ in radians per second-squared and $t$ in seconds. At time $t=0,$ the wheel has an angular velocity of $+2.0$ rad/s and an angular position of $+1.0$ rad. Write expressions for (a) the angular velocity (rad/s) and (b) the angular position (rad) as functions of time (s).
The angular acceleration of a wheel, as a function of time, is $\alpha=5.0 t^{2}-8.5 t,$ where $\alpha$ is in rad $/ \mathrm{s}^{2}$ and $t$ in seconds. If the wheel starts from rest $(\theta=0, \omega=0,$ at $t=0$ ), determine a formula for $(a)$ the angular velocity $\omega$ and $(b)$ the angular position $\theta,$ both as a function of time. (c) Evaluate $\omega$ and $\theta$ at $t=2.0 \mathrm{~s}$
(II) The angular acceleration of a wheel, as a function of time, is $\alpha=5.0 t^{2}-8.5 t,$ where $\alpha$ is in $\operatorname{rad} / \mathrm{s}^{2}$ and $t$ in seconds. If the wheel starts from rest $(\theta=0, \omega=0,$ at $t=0$ , determine a formula for $(a)$ the angular velocity $\omega$ and $(b)$ the angular position $\theta,$ both as a function of time. (c) Evaluate $\omega$ and $\theta$ at $t=2.0 \mathrm{s}$
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