Thursday, September 18 , 2012
AST101Y1 – Astronomy
Gravity and orbits; escape speed; Kepler's Third Law in detail
Textbook (pg. 114)
Chapter 4 – Making Sense of the Universe
4.1 Describing Motion: Examples from Daily Life
How do we describe motion?
Speed = The rate at which an object moves. Its units are distance divided by time, such as m/s or km/hr.
Velocity = The combination of speed and direction of motion; it can be stated as a speed in a particular
direction, such as 100 km/hr due north.
Acceleration =The rate at which an objects velocity changes. Its standard units are m/s .2
- Slowing occurs when acceleration is in a direction opposite to the motion. This is negative acceleration
which causes your velocity to decrease.
You don’t feel the effects of motion at constant velocity (e.g. airplane on a smooth flight)
Acceleration 2f Gravity = The acceleration of 2 falling object. On Earth, the acceleration of gravity, designated by
g, is 9.8 m/s , or less precisely, about 10 m/s .
- Means that an unsupported object’s downward velocity increases by about 10 m/s with each passing second
Momentum and Force
Speed, velocity, and acceleration describe how an individual object moves, but momentum and force result
from interactions between objects
Force = Anything that can cause a change in momentum.
Momentum = The product of an object’s mass and velocity. That is, momentum = mass × velocity.
- An object with greater mass will have greater momentum
- The mere presence of a force doesn’t always cause a change in momentum (there are always forces present
on Earth, e.g. gravity and the electromagnetic forces acting between atoms)
Net Force = The overall force to which an object responds; the net force is equal to the rate of change in the
object’s momentum, or equivalently to the object’s mass × acceleration.
A change in momentum occurs only when the net force is not zero. Changing an object’s momentum means
changing its velocity, as long as its mass remains constant. A net force that is not zero therefore causes an object
e.g. Planets are always accelerating as they orbit the Sun, because their direction of travel constantly changes as
they go around their orbits. Thus, some force, gravity, must be causing this acceleration.
Moving in Circles
Angular Momentum = Momentum attributable to rotation or revolution. The angular momentum of an object
moving in a circle of radius r is the product m × v × r.
- Any object spinning or moving along a curved path has angular momentum
e.g. Earth has angular momentum to its rotation (its rotational angular momentum) and to its orbit around
the Sun (its orbital angular momentum)
An object’s angular momentum can only be changed when a special type of force is applied to it
- Torque = A twisting force that can cause a change in an object’s angular momentum. The amount of torque
depends on how much force and where it is applied.
How is mass different from weight?
The distinction between mass and weight rarely matters on Earth, but does in the study of space
Mass = A measure of the amount of matter in an object.
Weight = The net force than an object applies to its surroundings; in the case of a stationary body on the surface
of Earth, it equals mass × acceleration of gravity. 2
Your mass depends only on the amount of matter in your body and is the same everywhere, but your weight can
vary because the forces acting on you can vary
e.g. Your mass would be the same on the Moon, but you would weigh less because of its weaker gravity
Free-Fall and Weightlessness in Space
Free-fall = The condition in which an object is falling without resistance; objects are weightless when in free-fall.
Gravity makes the Space Station and Shuttle orbit the Earth, but they stay in orbit because they are in a constant
state of free-fall which makes them weightless
4.2 Newton’s Laws of Motion
How did Newton change out view of the universe?
Newton’s discovery of gravity abolished Aristotle’s physics and signalled the birth of astrophysics
What are Newton’s three laws of motion?
Newton’s Laws of Motion = Three basic laws that describe how objects respond to forces.
1. An object moves at constant velocity if there is not net force acting upon it.
- Basically an object will remain in motion unless a force acts to stop it. Objects at rest (velocity = 0) tend to
remain at rest, and objects in motion tend to remain in motion with no change in either their speed or their
- This explains why interplanetary spacecraft don’t need fuel to keep travelling in space
2. Force = mass × acceleration (or F = ma)
- Because Jupiter is more massive than Earth, it exerts a stronger gravitational force on passing asteroids and
comets, and therefore sends them scattering with a greater acceleration
- Gravity is the force that keeps the planets accelerating toward the Sun
3. For any force, there is always an equal and opposite reaction force.
- Objects always attract each other through gravity Your body exerts the same amount of gravitational
force on the Earth that the Earth exerts on you, it just acts in the opposite direction. However, the same
force means a much greater acceleration for you than for Earth because your mass is so much smaller than
Earth’s which is why you fall toward Earth when you jump off a chair, rather than Earth falling toward you.
- Rocket propelled upward by a force equal and opposite to the force with which gas is expelled out the back
4.3 Conservation Laws of Astronomy
Newton’s laws reflect deeper aspects of nature known as conservation laws, three of which include the laws of
conservation of momentum, angular momentum, and energy
Why do objects move at constant velocity if no force acts on them?
Conservation of Momentum = The principle that, in the absence of net force, the total momentum of a system
- The total momentum off interacting objects cannot change as long as no external force is acting on them;
that is, their total momentum is conserved. An individual object can gain or lose momentum only if some
other object’s momentum changes by a precisely opposite amount.
From the perspective of this law, Newton’s first law makes perfect sense. When no net force acts on a subject,
there is no way for the object to transfer any momentum to or from another object. In the absence of a net
force, an object’s momentum must therefore remain unchanged, which means the object must continue to
move exactly as it has been moving.
What keeps the planet rotating and orbiting the Sun?
Conservation of Angular Momentum = The principle that, in the absence of net torque (twisting force), the total
angular momentum of a system remains constant.
- As long as there is no external torque, the total angular momentum of a set of interacting objects cannot
change. An individual object can change its angular momentum only by transferring some angular
momentum to or from another object. 3
Orbital Angular Momentum
Because there are no objects to give or take angular momentum from Earth as it orbits the Sun, Earth’s orbital
angular momentum must always stay the same, which explains two key facts about Earth’s orbit:
- Earth needs no fuel or push of any kind to keep orbiting the Sun. It will continue to do so as long as nothing
comes along to take angular momentum away.
- Because Earth’s angular momentum at any point in its orbit depends on the product of its speed and orbital
radius (distance from the Sun), Earth’s orbital speed must be faster when it is nearer to the Sun (and the
radius is smaller) and slower when it is farther from the Sun (and the radius is larger). This law thus basically
points out and tells us why Kepler’s second law is true.
Rotational Angular Momentum
As long as Earth isn’t transferring any of the angular momentum of its