Answer to Question #195320 in Physics for mazzu

Question

The rotating blade of a blender turns with a constant angular acceleration 1.50 [ rad/s^2 ]. a. How much time does it take to reach an angular velocity of 36.0 [ rad/s^2 ], starting from rest? b. Through how many revolutions does the blade turn in this time interval? c. What are the tangential acceleration and tangential speed of an ant sitting on the blade 10.0 cm from the axis of rotation during the end of this interval? d. What is the moment of inertia of the ant if its mass is 10.0 [g]? e. What is the rotational kinetic energy of the ant?

Answer 1

a)

$$\omega=\omega_0+\alpha t,$$ $$t=\dfrac{\omega-\omega_0}{\alpha},$$ $$t=\dfrac{36\ \dfrac{rad}{s}-0}{1.5\ \dfrac{rad}{s^2}}=24\ s.$$

b)

$$\theta=\omega_0t+\dfrac{1}{2}\alpha t^2,$$ $$\theta=0+\dfrac{1}{2}\cdot1.5\ \dfrac{rad}{s^2}\cdot(24\ s)^2=432\ rad,$$ $$n=\dfrac{\theta}{2\pi}=\dfrac{432\ rad}{2\pi}=69\ rev.$$

c)

$$a_t=r\alpha=0.1\ m\cdot1.5\ \dfrac{rad}{s^2}=0.15\ \dfrac{m}{s^2},$$ $$v_t=r\omega=0.1\ m\cdot36\ \dfrac{rad}{s}=3.6\ \dfrac{m}{s}.$$

d)

$$I_{ant}=m_{ant}r^2=0.01\ kg\cdot(0.1\ m)^2=1.0\cdot10^{-4}\ kg\cdot m^2.$$

e)

$$KE_{rot,ant}=\dfrac{1}{2}I\omega^2,$$ $$KE_{rot,ant}=\dfrac{1}{2}\cdot1.0\cdot10^{-4}\ kg\cdot m^2\cdot(36\ \dfrac{rad}{s})^2,$$ $$KE_{rot,ant}=0.065\ J.$$