**Chapter 8 Uniform circular motion and gravitation **

Many motions, such as the arc of a bird’s flight or the earth’s path around the sun are curved.

Uniform circular motion: motion in a circular path at constant speed.

**8.1 Rotation Angle and Angular Velocity **

Taken from Halliday

Taken from Halliday

Average angular velocity

Instantaneous angular velocity

Average angular acceleration

Instantaneous angular acceleration

Relating the linear and angular variables

The tangential component of the linear acceleration of the point

**8.2 Centripetal acceleration **

Taking the ratio of BC to BA in each triangle, we obtain

2

1 *v*

*v*
*v*= =

*r*
*r*
*v*

*v* = ∆

∆
*r* *r*
*v*= *v*∆

∆

*t*
*r*
*r*
*v*
*t*
*v*

∆

= ∆

∆

∆

*r*
*a* *v*
*t*
*v*

*t*

= 2

∆ =

∆

**8.3 Centripetal force **

*c*

*t* *F*

*r*
*mv*
*t* *ma*

*m* *v* = = =

∆

∆ ^{2}

Example

(a) Calculate the centripetal force exert on a 900 kg car that negotiates a 500 m radius curve at 25.0 m/s. (b) Assuming an unbanked curve, find the minimum static coefficient of friction between the tires and the road.

Sol:

a) (900 kg)(1.25 m/s^{2}) 1125 N 1130 N

2 = = = ≈

= _{c}

*c* *ma*

*r*
*mv*
*F*

b)

Friction is to the left, causing the car to turn; thus friction is the centripetal force in this case

*r*
*mv*
*mg*
*N*

*f*

*F*_{c}_{s}_{s}

= 2

=

=

= *µ* *µ*

13 . 0

2 =

= *rg*
*v*
*µ**s*

Banked curve

*r*
*N* *mv*

2

sin*θ* =

*mg*
*N*cos*θ* =

*r*
*mg* *mv*

2

cos sin =

*θ*
*θ*

*rg*
*v*^{2}
tan^{−}1

=
*θ*

**8.5 Newton’s universal law of gravitation **

**Gravitation near Earth’s surface **

Free-fall acceleration…

Taken from Halliday

**Ex **

**Taken from Halliday **

**Sol: **

**8.6 Kepler’s laws **

Special case: A circular orbit

**Exercises: 8.5, 8.31 **

**Chapter 9 Rotational Motion and Angular Momentum **

**9.2 Kinematics of Rotational Motion **

**9.3 Rotational Inertia **

**Calculating the rotational inertia **

**Many cases… .. **

**Ex **

**proof) **

**Angular Momentum **

**Conservation of Angular Momentum **

**Ex **

**Taken from Halliday **

**Exercises: 9.10, 9.20 **

**Chapter 10 Fluid Statics **

**10.1 What is a fluid **

**Fluids flow. More precisely, a fluid is a state of matter that yields to sideways or shearing forces. **

Solid and liquid

**10.2 Density **

**10.3 Pressure **

**A scalar!! **

**Taken from Halliday **

**10.4 Variation of pressure with depth in a fluid **

**Fluid at rest **

**Mercury barometer **

**Mercury Manometers **

By squeezing the bulb, the person making the measurement creates pressure, which is transmitted undiminished to both the main artery in the arm and the manometer. When this applied pressure exceeds blood pressure, blood flow below the cuff is cut off. The person making the measurement then slowly lowers the applied pressure and listens for blood flow to resume. Blood pressure sure pulsates because of the pumping action of the hear, reaching a maximum, called systolic pressure, and a minimum, called diastolic pressure, with each heartbeat.

**Pascal’s Principle **

**Taken from Halliday **

**10.7 Archimede’s Principle **

**11.1 Bernoulli’s Equation **

**Proof **

**Taken from Halliday **

**Exercises: 10.19, 10.42 **