Gravity, continued

Reading and Review

Gravitational Force at the Earth’s Surface The center of the Earth is one Earth radius away, so this is the distance we use: The acceleration of gravity decreases slowly with altitude......until altitude becomes comparable to the radius of the Earth. Then the decrease in the acceleration of gravity is much larger: g

In the Space Shuttle Astronauts in the space shuttle float because: a) they are so far from Earth that Earth’s gravity doesn’t act any more b) gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward c) while gravity is trying to pull them inward, they are trying to continue on a straight-line path d) their weight is reduced in space so the force of gravity is much weaker

Astronauts in the space shuttle float because they are in “free fall” around Earth, just like a satellite or the Moon. Again, it is gravity that provides the centripetal force that keeps them in circular motion. In the Space Shuttle Astronauts in the space shuttle float because: Follow-up: How weak is the value of g at an altitude of 300 km? a) they are so far from Earth that Earth’s gravity doesn’t act any more b) gravity’s force pulling them inward is cancelled by the centripetal force pushing them outward c) while gravity is trying to pull them inward, they are trying to continue on a straight-line path d) their weight is reduced in space so the force of gravity is much weaker

Satellite Motion: F G and a cp Consider a satellite in circular motion * : * not all satellite orbits are circular! Gravitational Attraction: Necessary centripetal acceleration: Does not depend on mass of the satellite! larger radius = smaller velocity smaller radius = larger velocity Relationship between F G and a cp will be important for many gravitational orbit problems

A geosynchronous satellite is one whose orbital period is equal to one day. If such a satellite is orbiting above the equator, it will be in a fixed position with respect to the ground. These satellites are used for communications and and weather forecasting.

Averting Disaster a) it’s in Earth’s gravitational field b) the net force on it is zero c) it is beyond the main pull of Earth’s gravity d) it’s being pulled by the Sun as well as by Earth e) none of the above The Moon does not crash into Earth because:

The Moon does not crash into Earth because of its high speed. If it stopped moving, it would, of course, fall directly into Earth. With its high speed, the Moon would fly off into space if it weren’t for gravity providing the centripetal force. Averting Disaster The Moon does not crash into Earth because: Follow-up: What happens to a satellite orbiting Earth as it slows? a) it’s in Earth’s gravitational field b) the net force on it is zero c) it is beyond the main pull of Earth’s gravity d) it’s being pulled by the Sun as well as by Earth e) none of the above

Two Satellites a) 1 / 8 b) ¼ c) ½ d) it’s the same e) 2 Two satellites A and B of the same mass are going around Earth in concentric orbits. The distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the centripetal force acting on B compared to that acting on A?

Using the Law of Gravitation: we find that the ratio is. Two Satellites a) 1 / 8 b) ¼ c) ½ d) it’s the same e) 2 Two satellites A and B of the same mass are going around Earth in concentric orbits. The distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the centripetal force acting on B compared to that acting on A? Note the 1/R 2 factor

Gravitational Potential Energy Gravitational potential energy, just like all other forms of energy, is a scalar. It therefore has no components; just a sign. Gravitational potential energy of an object of mass m a distance r from the Earth’s center: (U =0 at r -> infinity) Very close to the Earth’s surface, the gravitational potential increases linearly with altitude:

Energy Conservation (Remember: gravity is conservative force!) Total mechanical energy of an object of mass m a distance r from the center of the Earth: This confirms what we already know – as an object approaches the Earth, it moves faster and faster.

Escape Speed Escape speed: the initial speed a projectile must have in order to escape from the Earth’s gravity from total energy: If initial velocity = v e, then velocity at large distance goes to zero. If initial velocity is larger than v e, then there is non-zero total energy, and the kinetic energy is non-zero when the body has left the potential well

Maximum height vs. Launch speed Speed of a projectile as it leaves the Earth, for various launch speeds

Black holes If an object is sufficiently massive and sufficiently small, the escape speed will equal or exceed the speed of light – light itself will not be able to escape the surface. This is a black hole. The light itself has mass (in the mass/energy relationship of Einstein), or spacetime itself is curved

Gravity and light Light will be bent by any gravitational field; this can be seen when we view a distant galaxy beyond a closer galaxy cluster. This is called gravitational lensing, and many examples have been found.

General Relativity Previous effects examples Clocks run slower in gravitational field - must be taken into account for GPS systems to work - was missed in determining time for neutrinos to go from CERN to Gran Sasso!

Kepler’s Laws of Orbital Motion Johannes Kepler made detailed studies of the apparent motions of the planets over many years, and was able to formulate three empirical laws You already know about circular motion... circular motion is just a special case of elliptical motion 1. Planets follow elliptical orbits, with the Sun at one focus of the ellipse. Only force is central gravitational attraction - but for elliptical orbits this has both radial and tangential components Elliptical orbits are stable under inverse-square force law.

Kepler’s Laws of Orbital Motion 2. As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time. This is equivalent to conservation of angular momentum v Δt r

Kepler’s Laws of Orbital Motion 3. The period, T, of a planet increases as its mean distance from the Sun, r, raised to the 3/2 power. This can be shown to be a consequence of the inverse square form of the gravitational force. For a (near) circular orbit:

If you weigh yourself at the equator of Earth, would you get a bigger, smaller, or similar value than if you weigh yourself at one of the poles? a) bigger value b) smaller value c) same value Guess My Weight

If you weigh yourself at the equator of Earth, would you get a bigger, smaller, or similar value than if you weigh yourself at one of the poles? a) bigger value b) smaller value c) same value normal force you are in circular motion net inward forcenormal force must be slightly less than mg The weight that a scale reads is the normal force exerted by the floor (or the scale). At the equator, you are in circular motion, so there must be a net inward force toward Earth’s center. This means that the normal force must be slightly less than mg. So the scale would register something less than your actual weight. Guess My Weight

Earth and Moon I a) the Earth pulls harder on the Moon b) the Moon pulls harder on the Earth c) they pull on each other equally d) there is no force between the Earth and the Moon e) it depends upon where the Moon is in its orbit at that time Which is stronger, Earth’s pull on the Moon, or the Moon’s pull on Earth?

By Newton’s Third Law, the forces are equal and opposite. Earth and Moon I a) the Earth pulls harder on the Moon b) the Moon pulls harder on the Earth c) they pull on each other equally d) there is no force between the Earth and the Moon e) it depends upon where the Moon is in its orbit at that time Which is stronger, Earth’s pull on the Moon, or the Moon’s pull on Earth?

Review Newton’s law of universal gravitation force of gravity between any two objects is attractive, acting on a line between the two objects a sphere can be treated as a point mass at the sphere’s center above the surface of a planet (i.e., earth): Orbital motion problems balance force of gravity with necessary centripetal acceleration Gravitational potential energy chose U=0 point to be at r -> infinity total energy conserved escape velocity: K >= U

Force Vectors A planet of mass m is a distance d from Earth. Another planet of mass 2m is a distance 2d from Earth. Which force vector best represents the direction of the total gravitation force on Earth? a b c d e 2d2d d 2m2m m

a b c d e 2d2d d 2m2m m mgreater 2m The force of gravity on the Earth due to m is greater than the force due to 2m, which means that the force component pointing down in the figure is greater than the component pointing to the right. A planet of mass m is a distance d from Earth. Another planet of mass 2m is a distance 2d from Earth. Which force vector best represents the direction of the total gravitation force on Earth? Force Vectors 2m2d GMm / d 2 F 2m = GM E (2m) / (2d) 2 = GMm / d 2 md GMm / d 2 F m = GM E m / d 2 = GMm / d 2

If you weigh yourself at the equator of Earth, would you get a bigger, smaller, or similar value than if you weigh yourself at one of the poles? a) bigger value b) smaller value c) same value Guess My Weight

If you weigh yourself at the equator of Earth, would you get a bigger, smaller, or similar value than if you weigh yourself at one of the poles? a) bigger value b) smaller value c) same value normal force you are in circular motion net inward forcenormal force must be slightly less than mg The weight that a scale reads is the normal force exerted by the floor (or the scale). At the equator, you are in circular motion, so there must be a net inward force toward Earth’s center. This means that the normal force must be slightly less than mg. So the scale would register something less than your actual weight. Guess My Weight

Oscillations about Equilibrium

Periodic Motion Period: time required for one cycle of periodic motion Frequency: number of oscillations per unit time This unit is called the Hertz:

Simple Harmonic Motion A spring exerts a restoring force that is proportional to the displacement from equilibrium: Displaced, at rest Moving, past equilibrium point Displaced, at rest Moving, past equilibrium point Displaced, at rest This is called “Simple Harmonic Motion”

Simple Harmonic Motion A mass on a spring has a displacement as a function of time that is a sine or cosine curve: Here, A is called the amplitude of the motion.

Simple Harmonic Motion If we call the period of the motion T – this is the time to complete one full cycle – we can write the position as a function of time: The position at time t +T is the same as the position at time t, as we would expect. TimePosition t =0x = A t=Tx = Acos(2π) = A t = T/2x = Acos(π) = -A t = T/4x = Acos(π/2) = 0 T

Sine vs Cosine x at t=0 : A x at t=0 : 0 v at t=0 : 0 v at t=0 : >0

Harmonic Motion I a) 0 b) A/2 c) A d) 2A e) 4A A mass on a spring in SHM has amplitude A and period T. What is the total distance traveled by the mass after a time interval T?

Harmonic Motion I a) 0 b) A/2 c) A d) 2A e) 4A A mass on a spring in SHM has amplitude A and period T. What is the total distance traveled by the mass after a time interval T? time interval T complete oscillationThe distance it covers is A + A + A + A = (4A). In the time interval T (the period), the mass goes through one complete oscillation back to the starting point. The distance it covers is A + A + A + A = (4A).

A mass on a spring in SHM has amplitude A and period T. What is the net displacement of the mass after a time interval T? a) 0 b) A/2 c) A d) 2A e) 4A Harmonic Motion II

A mass on a spring in SHM has amplitude A and period T. What is the net displacement of the mass after a time interval T? a) 0 b) A/2 c) A d) 2A e) 4A The displacement is  x = x 2 – x 1. Because the initial and final positions of the mass are the same (it ends up back at its original position), then the displacement is zero. Harmonic Motion II Follow-up: What is the net displacement after a half of a period?

The Pendulum A simple pendulum consists of a mass m (of negligible size) suspended by a string or rod of length L (and negligible mass). The angle it makes with the vertical varies with time as a sine or cosine.

How a pendulum is like the mass on a spring Looking at the forces on the pendulum bob, we see that the restoring force is proportional to sin θ The restoring force for a spring is proportional to the displacement This is the condition for simple harmonic motion

Approximation for sin θ However, for small angles, sin θ and θ are approximately equal. θ (deg)θ (rad)sin(θ)

Pendulum for small angles = simple harmonic for small angles of the pendulum bob, the restoring force is proportional to θ F = -mg θ = -mg s / L The restoring force for a spring is proportional to the displacement F= -kx So: the motion of the angle of the pendulum is the same as the motion for the mass on a spring, with k mg/L

Uniform Circular Motion and Simple Harmonic Motion An object in simple harmonic motion has the same motion as one component of an object in uniform circular motion Assume oscillation of mass on the spring has the same period T as the circular motion of the peg on the “record player”

Here, the object in circular motion has an angular speed of: where T is the period of motion of the object in simple harmonic motion. The position as a function of time:... just like the simple harmonic motion! Position of Peg in Circular Motion

Velocity of Peg in Circular Motion Linear speed v = Aω x component:

Acceleration of Peg in Circular Motion Linear acceleration a = Aω 2 x component:

Summary of Simple Harmonic Motion The position as a function of time: The angular frequency: From this comparison with circular motion, we can see: The velocity as a function of time: The acceleration as a function of time:

Speed and Acceleration a) x = A b) x > 0 but x < A c) x = 0 d) x < 0 e) none of the above A mass on a spring in SHM has amplitude A and period T. At what point in the motion is v = 0 and a = 0 simultaneously?

Speed and Acceleration a) x = A b) x > 0 but x < A c) x = 0 d) x < 0 e) none of the above A mass on a spring in SHM has amplitude A and period T. At what point in the motion is v = 0 and a = 0 simultaneously? If both v and a were zero at the same time, the mass would be at rest and stay at rest!NO point va If both v and a were zero at the same time, the mass would be at rest and stay at rest! Thus, there is NO point at which both v and a are both zero at the same time. Follow-up: Where is acceleration a maximum?

The Period of a Mass on a Spring For the mass on a spring: Substituting the time dependencies of a and x gives and the period is:

The Period of a Mass on a Spring

Vertical Spring What if the mass hangs from a vertical spring? F s =kx d x=0 W=mg new equilibrium position: x= -d = -mg/k total force as a function of x: x=-d with Looks like the same spring, with a different equilibrium position (x’=0 -> x = -d) Simple harmonic motion is unchanged from the horizontal case!

Energy Conservation in Oscillatory Motion In an ideal system with no nonconservative forces, the total mechanical energy is conserved. For a mass on a spring: Since we know the position and velocity as functions of time, we can find the maximum kinetic and potential energies:

Energy Conservation in Oscillatory Motion As a function of time, So the total energy is constant; as the kinetic energy increases, the potential energy decreases, and vice versa. 0

Period of a Pendulum A pendulum is like the mass on a spring, with k=mg/L Therefore, we find that the period of a pendulum depends only on the length of the string:

Physical Pendula A physical pendulum is a solid mass that oscillates around its center of mass, but cannot be modeled as a point mass suspended by a massless string. Examples:

Period of a Physical Pendulum In this case, it can be shown that the period depends on the moment of inertia: Substituting the moment of inertia of a point mass a distance l from the axis of rotation gives, as expected:

Damped Oscillations In most physical situations, there is a nonconservative force of some sort, which will tend to decrease the amplitude of the oscillation, and which is typically proportional to the speed: This causes the amplitude to decrease exponentially with time:

Damped Oscillations This exponential decrease is shown in the figure: “underdamped” means that there is more than one oscillation

Damped Oscillations The previous image shows a system that is underdamped – it goes through multiple oscillations before coming to rest. A critically damped system is one that relaxes back to the equilibrium position without oscillating and in minimum time; an overdamped system will also not oscillate but is damped so heavily that it takes longer to reach equilibrium.

Driven Oscillations and Resonance An oscillation can be driven by an oscillating driving force; the frequency of the driving force may or may not be the same as the natural frequency of the system.

Driven Oscillations and Resonance If the driving frequency is close to the natural frequency, the amplitude can become quite large, especially if the damping is small. This is called resonance.

A mass on a spring oscillates in simple harmonic motion with amplitude A. If the mass is doubled, but the amplitude is not changed, what will happen to the total energy of the system? a) total energy will increase b) total energy will not change c) total energy will decrease Energy in SHM I

A mass on a spring oscillates in simple harmonic motion with amplitude A. If the mass is doubled, but the amplitude is not changed, what will happen to the total energy of the system? a) total energy will increase b) total energy will not change c) total energy will decrease The total energy is equal to the initial value of the elastic potential energy, which is PE s = kA 2. This does not depend on mass, so a change in mass will not affect the energy of the system. Energy in SHM I Follow-up: What happens if you double the amplitude?

A mass oscillates on a vertical spring with period T. If the whole setup is taken to the Moon, how does the period change? a) period will increase b) period will not change c) period will decrease Spring on the Moon

A mass oscillates on a vertical spring with period T. If the whole setup is taken to the Moon, how does the period change? a) period will increase b) period will not change c) period will decrease The period of simple harmonic motion depends only on the mass and the spring constant and does not depend on the acceleration due to gravity. By going to the Moon, the value of g has been reduced, but that does not affect the period of the oscillating mass – spring system. Spring on the Moon