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Center of mass

Rotational to Linear relationships

Relationship to other angular terms

Kinematics

A skater holds her arms outstretched as she spins at 80.0 rpm . What is the speed of her hands if they are 140 cm apart?

5.86 m/s

**A high speed drill reaches 2000 rpm in 0.5 s**

What is the drill's angular acceleration?

Through how many revolutions does it turn during this first 0.5 s?

What is the drill's angular acceleration?

Through how many revolutions does it turn during this first 0.5 s?

**419 rad/s^2**

**8.3 rev**

An 18 cm long bicycle crank arm, with a pedal at one end is attached to a 20 cm diameter sprocket, the toothed disk around which the chain moves. A cyclist riding this bike increases her pedaling rate from 60 rpm to 90 rpm in 10 s.

A. What is the tangential acceleration of the pedal?

B. What length of chain passes over the top of the sprocket during this interval?

.057 m/s^2

7.9 m

The three masses shown are connected by massless rigid rods. What are the coordinates of the center of mass?

100 g

200 g

300 g

10 cm

10 cm

(6.67, 5.0)

A turbine blade rotates with angular velocity ω(t) = 6.00 rad/s - 2.10 rad/s^3 t 2. What is the angular acceleration of the blade at t = 5.6 s?

-23.5 rad/s^2

A 3.45-kg centrifuge takes 100 s to spin up from rest to its final angular speed with constant angular acceleration. A point located 8.00 cm from the axis of rotation of the centrifuge moves with a speed of 150 m/s when the centrifuge is at full speed.

(a) What is the angular acceleration (in rad/s2) of the centrifuge as it spins up?

(b) How many revolutions does the centrifuge make as it goes from rest to its final angular speed?

14900 rev

18.8 rad/s^2

A uniform piece of wire, 20 cm long, is bent in a right angle in the center to give it an L-shape. How far from the bend is the center of mass of the bent wire?

3.54 cm

A long thin rod of length L has a linear density λ(x)=Ax where x is the distance from the left end of the rod.

(a) How far is the center of mass of the rod from the left end of the rod?

(b) What is the mass of the rod?

AL^2/2

2/3 L

An 800 g steel plate has the shape of the isosceles triangle shown in the figure

What is the x-coordinates of the center of mass?

Hint: Divide the triangle into vertical strips of width dx, then relate the mass dm of a strip at position x to the values of x and dx.

20 cm

**Chapter 12 Practice 1**

"What does it mean that every point on a solid rotating disk turns through the same angle in time?"

"What is the r in most of the equations?"

"What is the mass-weighted center"

"I still don´t undersand how to calculate the moment of inertia." (tomorrow)

Consider the case that the string tied to the block is wrapped around the outside of the wheel, which has a radius rA. Find ωA, the angular speed of the wheel after the block has fallen a distance h, for this case.

Express ωA in terms of m, g, h, rA, and I.