Circular Motion. Problem 1 A loop-the-loop machine has radius r of 18m. a.Calculate the minimum...
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Transcript of Circular Motion. Problem 1 A loop-the-loop machine has radius r of 18m. a.Calculate the minimum...
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