Circular Motion

Problem 1 A loop-the-loop machine has radius r of 18m.

a.Calculate the minimum speed with which a cart must enter the loop so that it does not fall off at the highest point.

b.Predict the speed at the top in this case.

Problem 2

In an amusement park ride a cart of mass 300kg and carrying four passengers each mass 60kg is dropped from a vertical height of 120 m along a frictionless path that leads into a loop-the-loop machine of radius 30m. The cart then enters a straight stretch from A to C where friction brings it to rest after a distance of 40 m

a. Determine the velocity of the cart at A.

b. Calculate the reaction force from the seat of the cart onto a passenger at B.

c. Determine the acceleration experienced by the cart from A to C (assumed constant)

The Law of GravitationTOPIC 6

Topic 6: Fields and forces

State Newton’s universal law of gravitation.

Students should be aware that themasses in the force law are pointmasses. The force between twospherical masses whose separationis large compared to their radiiis the same as if the two sphereswere point masses with theirmasses concentrated at the centersof the spheres.

6.1 Gravitational force and field

Must be true from Newton’s 3rd Law

Earth exerts a downward force on you, & you exert an upward force on Earth. When there is such a large difference in the 2 masses, the reaction force (force

you exert on the Earth) is undetectable, but for 2 objects with masses closer in size to each other, it can be significant.

The gravitational force one body exerts on a 2nd body , is directed toward the first body, and is equal and opposite to the force exerted by the second body on the first

Newton’s Universal Law of Gravitation

Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between them.

F12 = -F21 [(m1m2)/r2]

Direction of this force: Along the line joining the 2 masses

Comments

F12 Force exerted by particle 1 on particle 2

F21 Force exerted by particle 2 on particle 1

This tells us that the forces form a Newton’s 3rd Law action-reaction pair, as expected.

The negative sign in the above vector equation tells us thatparticle 2 is attracted toward particle 1

F21 = - F12

More Comments

Gravity is a field force that always exists between 2 masses, regardless of the medium between them.

The gravitational force decreases rapidly as the distance between the 2 masses increases› This is an obvious consequence of the

inverse square law

Example : Spacecraft at 2rE

• Earth Radius: rE = 6320 km

Earth Mass: ME = 5.98 1024 kg

FG = G(mME/r2)

Mass of the Space craft m• At surface r = rE

FG = weight

or mg = G[mME/(rE)2]

• At r = 2rE

FG = G[mME/(2rE)2]

or (¼)mg = 4900 N

• A spacecraft at an altitude of twice the Earth radius

Example : Force on the Moon

Find the net force on the

Moon due to the gravitational

attraction of both the Earth &

the Sun, assuming they are at

right angles to each other.

ME = 5.99 1024kg

MM = 7.35 1022kg

MS = 1.99 1030 kg

rME = 3.85 108 m

rMS = 1.5 1011 m

F = FME + FMS

F = FME + FMS

(vector sum)

FME = G [(MMME)/ (rME)2]

= 1.99 1020 N

FMS = G [(MMMS)/(rMS)2]

= 4.34 1020 N

F = [ (FME)2 + (FMS)2]

= 4.77 1020 N

tan(θ) = 1.99/4.34

θ = 24.6º

Gravitational Field Strength

Consider a man on the Earth:

Man’s weight = mgBUT we know that this is equal to his gravitational attraction, so…

GMm = mg

r2

GM = g

r2

Therefore:

(this is a vector quantity)

Derive an expression for gravitational field strength at the surface of a planet, assuming that all its mass is concentrated at its centre.

Force per unit point mass

Prob. 3

Estimate the force between the Sun and the Earth.

Prob.4

Determine the acceleration of free fall (the gravitational field strength) on a planet 10 times as massive as the Earth and with a radius 20 times as large.

Orbits and Gravity

Orbital Equation

Predicts the speed of the satellite at a particular radius

Angular speed

Orbital Period

Kepler’s Third Law