Reading Quiz 1. Which is true? The gravitational force between two particles ___ 1. can be shielded by the presence of an intervening mass. ___ 2. is inversely.

Reading Quiz1. Which is true? The gravitational

force between two particles___ 1. can be shielded by the presence

of an intervening mass.___ 2. is inversely proportional to the

distance between the particles.___ 3. obeys the law of superposition.___ 4. is independent of the distance

between the particles.

2. The gravitational constant G is___ 1. equal to g at the surface of Earth.___ 2. different on the Moon than on Earth.___ 3. obtained by measuring the speed of

falling objects having different masses.___ 4. none of the above

3. Which is one of Kepler’s laws?___ 1. The gravitational attraction of Earth

and the Sun provides a centripetal

acceleration explaining Earth’s orbit.

___ 2. The gravitational and inertial masses of an object are equivalent.

___ 3. The radial line segment from the Sun to a planet sweeps out equal areas in equal time intervals.

4. Which term was not introduced in

today’s reading assignment?___ 1. escape velocity___ 2. perihelion___ 3. gravitational mass___ 4. Hubble’s constant

Circular Motion• Motion with constant speed in a circle - is

the particle accelerated? Why?

• Centripetal acceleration - acceleration due to change in direction of velocity vector:

(magnitude)

• Direction - radially inward.

• Centripetal force: force needed to provide centripetal acceleration - Newton’s 2nd Law gives magnitude and direction of force: radially inward

2R

va r

2R

vF m r

Conceptual Questions1) The game of tetherball is played with a ball tied to

a pole with a string. When the ball is struck, it whirls around the pole as shown. At this moment, in what direction is the acceleration of the ball, and what causes the acceleration?

____ a) up ____ a) gravity

____ b) down ____ b) gravity

____ c) right ____ c) string

____ d) left ____ d) string

____ e) not accelerated ____ e) not needed

2) Suppose you are in a car moving with a

constant speed along a straight road. You have a pendulum that is hanging freely. If the car now approaches a curve and makes a left turn, which way will the bob move?

____ a) left

____ b) right

____ c) up

____ d) forward

____ e) backward

3) Suppose you have a stone tied to a string, and you swing the stone around (clockwise when viewed from above) in an almost horizontal circle. If the string breaks just when the stone is on the north point of the compass, how does the stone move subsequently?

____ a) northward, projectile motion

____ b) southward, projectile motion

____ c) eastward, projectile motion

____ d) westward, projectile motion

____ e) south-east, projectile motion

____ f) north-east, projectile motion

4) Consider a particle moving with constant

speed such that its acceleration of constant magnitude is always perpendicular to its velocity.

____ a) It is moving in a straight line.

____ b) It is moving in a circle.

____ c) It is moving in a parabola.

____d) None of the above is definitely true all of the time.

Quantitative Questions1) A flat puck (mass M) is rotated in a circle on a

frictionless air hockey tabletop, and is held in this orbit by a light cord which is connected to a dangling mass (mass m) through the central hole as shown in the figure below. Find the speed of the puck in terms of the given quantities.

2) At what minimum speed must a roller coaster be traveling when upside down at the top of a circle if the passengers are not to fall out? Assume the radius of curvature is 8.6 m.

3) A device for training astronauts and jet fighter pilots is designed to rotate the trainee in a horizontal circle of radius 10.0 m. If the force felt by the trainee is 7.75 times her own weight, how fast is she rotating? How many revolutions per second is that?

Banking & Centrifuge• Why do cars skid (in what direction) when

rounding a curve? What provides the force?

• Banking - angling of surface so normal force provides centripetal force. Examples: banked roads, velodromes, airplanes & birds.

• Centrifuge - effective “gravity” that can be used to separate materials with slightly different characteristics: washer.

• Non-uniform Circular Motion: centripetal and tangential accelerations.

Conceptual Problems

1) If the bank of a road is designed for a speed of 50 mph, how does the car tend to slip at a speed of 60 mph?

____ a) not at all

____ b) inwards towards the center of curvature

____ c) outwards away from the center

What (if anything) prevents this from happening?

2) Artificial gravity is produced in a toroidal space station by having it spinning about its axis. What direction will “up” be for the astronauts?

____ a) radially outward

____ b) radially inward

____ c) straight towards the earth

____ d) it is impossible to produce artificial gravity

3) Two particles of the same size and shape

but different densities are suspended in a liquid. This is put into an ultracentrifuge. Which particle will reach the bottom first?

____ a) the denser one

____ b) the less dense one

____ c) they both reach at the same time

____ d) they do not move

Quantitative Problems1) For a car traveling with speed v around a

curve of radius r, determine a formula for the angle at which a road should be banked so that no friction is required.

2) An airplane is flying in a horizontal circle

at a speed of 480 km/h. If its wings are tilted 40 to the horizontal, what is the radius of the circle in which the plane is flying? Assume that the required force is provided entirely by an “aerodynamic lift” that is perpendicular to the wing surface.

Newton’s Gravitation• Every particle in the universe attracts every

other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along the line joining the two particles: G = universal gravitation constant mA = mass of first particle mB = mass of second particle r = distance between the two particles

grav 2BA

m mF G

r

• Experimentally verified.

• Responsible for the motion of all the heavenly bodies. On the cosmic scale, this is the dominant force.

• Why is acceleration near the surface of the earth equal to 9.8 m/s2?

Conceptual Question

Where is the gravitational attraction of the earth greatest?

____ a) at the Poles

____ b) at the equator

____ c) same all over the earth

Quantitative Problems1) Calculate the acceleration due to gravity on the

moon. The moon’s radius is about 1.74 x 106 m and its mass is 7.35 x 1022 kg.

2) Determine the mass of the sun. Take the distance from the sun to be 1.5 x 1011 m.

3) A geo-synchronous satellite is one that stays above the same point on the equator of the earth. Such satellites are used for purposes as cable TV transmission, for weather forecasting, and as communication relays. What is the height above the earth’s surface such a satellite must orbit? Do lower orbit satellites move faster or slower?

Kepler’s Laws of Planetary Motion• First Law: The path of

each planet about the sun is an ellipse with the sun at one focus

• Second Law: Each planet moves so that an

imaginary line drawn from the sun to the planet sweeps out equal areas in equal periods of time.

• Third Law: The ratio of the squares of the

periods of any two planets revolving about the sun is equal to the ratio of the cubes of their mean distances from the sun:

2 3

A ABB

T rrT

Kepler’s Law of Periods for the Solar System

Conceptual Problems

1) According to Kepler’s third law, the time needed for a planet to go around the sun

____ a) depends on its mass

____ b) depends on the average radius of orbit

____ c) depends on its speed of rotation

____ d) is the same for all the planets

2) The speed of a planet in its elliptical orbit around the sun

____ a) is constant

____ b) is highest when it is closest to the sun

____ c) is lowest when it is closest to the sun

____ d) varies, but not with respect to its distance from the sun

3) Satellite 1 is in a certain circular orbit

about a planet, while satellite 2 is in a larger circular orbit. Which satellite has the longer period and greater speed?

____ a) satellite 1 and satellite 1

____ b) satellite 1 and satellite 2

____ c) satellite 2 and satellite 1

____ d) satellite 2 and satellite 2

____ e) they have the same period and speed