CHAPTER 6CIRCULAR MOTION AND GRAVITATION

Goals for Chapter 6

• To understand the dynamics of circular motion.

• To study the unique application of circular motion as it applies to Newton’s Law of Gravitation.

• To study the motion of objects in orbit as a special application of Newton’s Law of Gravitation.

Uniform circular motion is due to a centripetal accelerationThis aceleration is always pointing to the centerThis aceleration is dur to a net force

• Period = the time for one revolution

1) Circular motion in horizontal plane: - flat curve - banked curve - rotating object2) Circular motion in vertical plane

Rounding a flat curve

• The centripetal force coming only from tire friction.

Rounding a banked curve

• The centripetal force comes from friction and a component of force from the car’s mass

A tetherball problem

Dynamics of a Ferris Wheel

HOMEWORK CH6

3; 5; 8; 13; 15; 19; 22; 29; 36; 40

• Each stride is taken as one in a series of arcs

Spherically symmetric objects interact

gravitationally as though all the mass of each were concentrated at its center

A diagram of gravitational force

Newton’s Law of Gravitation

• Always attractive.• Directly proportional to the masses involved.• Inversely proportional to the square of the

separation between the masses.• Masses must be large to bring Fg to a size

even close to humanly perceptible forces.

•Use Newton’s Law of Universal Gravitation with the specific masses and separation.

•The slight attraction of the masses causes a nearly imperceptible rotation of the string supporting the masses connected to the mirror.

•Use of the laser allows a point many meters away to move through measurable distances as the angle allows the initial and final positions to diverge.

Cavendish Balance

WEIGHT

Gravitational force falls off quickly• If either m1 or m2 are small, the force decreases quickly enough for humans to notice.

What happens when velocity rises? • Eventually, Fg balances and you have orbit.

• When v is large enough, you achieve escape velocity.

Satellite Motion

The principle governing the motion of the satellite is Newton’s second law; the force is F, and the acceleration is v2/r, so the equation Fnet = ma becomes

GmmE/r 2 = mv 2/r

v = GmE/r

Larger orbits correspond to slower speeds and longer periods.

We want to place a satellite into circular orbit 300km above the earth surface.

What speed, period and radial acceleration it must have?

A 320 kg satellite experiences a gravitational force of 800 N. What is the radius of the of the satellite’s orbit? What is its altitude?F = GmEmS/r 2

r 2 = GmEmS/ F

r 2 = (6.67 x 10 -11 N.m2/kg2) (5.98 x 10 24 kg) (320 kg ) / 800 N

r 2 = 1.595 x 1014 m2

r = 1.26 x 107 m

Altitude = 1.26 x 107 m – radius of the Earth Altitude = 1.26 x 107 m – 0.637 x 107 = 0.623 x 107 m