Circular Motion

Problem 1 A loop-the-loop machine has radius r of 18m. Calculate the minimum speed with which a cart must enter the loop so that it does not fall off at the highest point. 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 Determine the velocity of the cart at A. Calculate the reaction force from the seat of the cart onto a passenger at B. Determine the acceleration experienced by the cart from A to C (assumed constant)

The Law of Gravitation Topic 6

Topic 6: Fields and forces 6.1 Gravitational force and field State Newton’s universal law of gravitation. Students should be aware that the masses in the force law are point masses. The force between two spherical masses whose separation is large compared to their radii is the same as if the two spheres were point masses with their masses concentrated at the centers of the spheres.

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 Figure 6-2. Caption: The gravitational force one object exerts on a second object is directed toward the first object, and is equal and opposite to the force exerted by the second object on the first.  Must be true from Newton’s 3rd Law

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 F21 = - F12 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 that particle 2 is attracted toward particle 1

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 A spacecraft at an altitude of twice the Earth radius 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

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 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 Force per unit point mass Derive an expression for gravitational field strength at the surface of a planet, assuming that all its mass is concentrated at its centre. Man’s weight = mg BUT we know that this is equal to his gravitational attraction, so… GMm = mg r2 Therefore: GM = g r2 (this is a vector quantity) Consider a man on the Earth:

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