**Rotational Dynamics**

This section is interesting, as all your kinematic equations can be converted
into rotational terms.

Theta = radians.

w = angular velocity.

alpha = angular acceleration.

360 degrees = 2pi radians.

w = w0 + alpha(t)

w^2 = w0^2 + 2(alpha)(theta) **Use when time is not given.**

Theta = w0t + 1/2(alpha)t^2 **Use when time is given.**

2pi(f)=w. **F is in revs/second.**

I = moment of inertia **(equation depends on object shape).**

L = angular momentum = Iw.

Torque = r_|_F = I(alpha)

v = rw

a_tan = r(alpha)

a_radial = w^2r

** 1.** There is a difference between translation and rotational
motion. In translation motion, objects move without rotating. Rotational motion
on the other hand, is when all points in the rigid body move in circles.

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2.** A radian is defined as the angle subtended by an arc whose length
is equal to the radius. In a circle, there are 2pi radians, so 2pi radians is
equivalent to 360 degrees. Theta = length of arc/radius.

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**Problem 1.
**A tire is accelerates east from rest at 1.0m/s^2. If the diameter of
the wheel is .68m, what is the velocity tangent after 3 seconds at the top of
the tire if a point at the bottom of the tire is considered to be at rest?

Calculate angular acceleration.

a = r(alpha).

1.0m/s^2 = .34m(alpha).

Calculate angular velocity.

w=w0+alpha(t)

w = 0 + (2.94rads/s^2)(3)

Now, consider a circle with its center at the bottom of the tire with a radius of the tire's diameter. Since angular velocity is the same no matter what the radius is, it can be used to determine the tangential velocity.

v_tan = rw

v_tan = (0.68m)(8.82rads/s)

**Problem 2
**A ducky is watching a chicken crawling on the still blade of a ceiling
fan when he decides to turn on the fan and watch the chicken go for a ride.
If the chicken sits on the fan blade at a distance of 0.80m from the center
of the fan, and turns with a frequency of 1.2Hz, what is the linear speed of
the chicken?

Calculate the angular velocity of the chicken, I mean, fan. *(inside joke)*

2pi(f) = w.

2pi(1.2revs/second) = w

7.54 rads/second = w

Calculate the linear velocity of the chicken.

v = rw

v = (0.80m)(7.54rads/second)

*v = 6.03m/s*

**Problem 3** ( hi rina! )

The moon orbits the earth so that the same side always faces the earth. Determine the ratio of its spin angular momentum to its orbital angular momentum, treating the moon as a particle orbiting the earth.

The moon spins about its axis once every revolution about the earth, this is because the problem states that the same side always faces the earth.

As a result, the angular velocity, omega, is the same for both angular momentums since the time to revolve about the earth is the same as the time to spin about its axis.

Angular momentum is given by L = Iw.

The moon can be considered as a solid sphere, so I = 2/5M_moon*R_moon^2. We use this I in the spin angular momentum, L_spin.

For the orbital angular momentum, we can consider the earth-moon system as a hoop (moon as particle), since the moon revolves about the earth: I = M_moon*R_moon-earth^2 is used in L_orbital.

The ratio is then L_spin/L_orbital = ((2/5)M_moon*R_moon^2)/(M_moon*R_moon-earth^2).

*Note: the omegas canceled.