Question

A ball is spun with angular acceleration $\alpha=6 t ^{2}-2 t$ where $t$ is in second and $\alpha$ is in rads $^{-2} .$ At $t=0$, the ball has angular velocity of $10$ rads $^{-1}$ and angular position of $4\, rad$. The most appropriate expression for the angular position of the ball is:

• A. $\frac{3}{2} t^{4}-t^{2}+10 t$
• B. $\frac{t^{4}}{2}-\frac{t^{3}}{3}+10 t+4$
• C. $\frac{2 t^{4}}{3}-\frac{t^{3}}{6}+10 t+12$
• D. $2 t^{4}-\frac{t^{3}}{2}+5 t+4$

Answer 1

Answer: (B)

$\frac{d w}{d t}=6 t^{2}-2 t$
$\int\limits_{10}^{w} d w=2 t^{3}-t^{2}$
$w=10+2 t^{3}-t^{2}$
$\frac{d \theta}{d t}=10+2 t^{3}-t^{2}$
$\int\limits_{4}^{\theta} d \theta=10+2 t^{3}-t^{2}$
$\theta=4+10 t+\frac{t^{4}}{2}-\frac{t^{3}}{3}$