Download Newton*s second law of motion - crypt

11 OUT INTO SPACE
Rhythms of the heavens
• Review knowledge and
understanding of
planetary motions
• Learn Kepler’s laws of
planetary motion
• Analyse planetary orbit
data to determine
Kepler’s 3rd law
empirically
Q1. What is the significance of each of the following:
23 hours 56 min ; 24 hours ; 24 hours 49 min ;
28 days
; 365.25 days
Q2. Why do celestial bodies move across the sky from East to West each night? Are
there any that don’t? Why not?
Q3. Why are the planets sometimes referred to as “wandering stars”?
Q4. Why does Mars show retrograde motion when tracked over several months?
Q5. Why do we see different constellations at
different times of year?
Q6. Why is Halley’s comet only seen once
every 76 years?
Q7. Why do we only see one side of the Moon
from Earth?
Q1. Use Kepler’s 3rd law (T2 = constant x R3) to predict the time it takes
Saturn to go once round the Sun:
Venus
Earth
Mars
Jupiter
Saturn
Mean distance
from Sun (AU)
Orbital
period (years)
0.723
1.000
1.520
5.20
9.54
0.616
1.000
1.88
11.9
?
Q2. Can you justify Kepler’s 2nd Law (p36) in terms of conservation of
energy? (Hint: For a planet orbiting the Sun, the sum of KE + GPE will be
constant)
Uniform circular motion
• Explain circular
motion in terms of
centripetal force,
identifying its origin
• Derive and use the
equations of circular
motion
Q1. Calculate the angular velocity (in Radians s-1) for the following examples of
uniform circular motion:
(a) The seconds hand on a watch
(b) The minute hand on a watch
(c) The hour hand on a watch
Q2. Calculate the centripetal acceleration of a point on the equator of the
Earth due to the rotation of the Earth. Data: radius of Earth = 6.36 x 106 m
Q3. The measured acceleration due to gravity at the Equator is about 9.78
ms-2, while at the poles it is 9.83 ms-2. Explain these observations in light of
your answer to Q2.
Q4. You swing a bucket of water, mass 5 kg, on a 0.9 m long string in a vertical
circle, so that the bucket completes two complete revolutions in 3 seconds.
(a) Calculate the centripetal acceleration and centripetal force acting on the
bucket.
(b) What is the tension in the string when the bucket is at (i) the top of the circle
and (ii) the bottom of the circle?
A feeling of “weightlessness” will arise if the
centripetal force is equal to the weight of the
object.
If a “vomit comet” follows a circular arc of
radius 1000 m to achieve “weightlessness” of
the occupants, what is the required speed?
Newton’s Law of Universal Gravitation
• Understand Newton’s
reasoning in deriving
NLUG
• Derive Kepler’s 3rd law
from a consideration of
NLUG and centripetal
force
• Investigate stability of
orbital systems
The Moon takes about 28 days to orbit the Earth. The Moon is
about 384 000 km from the centre of the Earth.
Q1. Calculate the angular velocity (in Radians s-1) of the Moon as it orbits the Earth.
Q2. Calculate the centripetal acceleration of the Moon (ms-2).
Q3. What provides the centripetal force that keeps the Moon orbiting the Earth?
Q4. What is the acceleration of an object at the surface of the Earth (ms-2)?
Q5. What provides the force that causes this acceleration?
Q6. Newton’s hypothesis was that it was the Earth’s gravity, “diluted” with distance, that
provided the force to keep the Moon in orbit. Complete the table below and test to
see how g decreases with distance.
(Hint: Test g = k/dn, where k is a constant and n is an integer, for different values of n.
Radius of Earth = 6360 km)
Location
Surface of Earth
Moon
Distance from
Earth’s centre (km)
g (ms-2)
NLUG proves Kepler #3 !
1. Write down an expression for
the gravitational force
between the Sun and a planet
(Eqn.1)
2. Write down an expression for
the angular velocity of a
planet in terms of its period T
(time to go round Sun) (Eqn.2)
3. Write down an expression for
the centripetal force on the
planet in terms of ω. (Eqn.3)
4. Substitute Eqn.2 into Eqn.3 to
get Eqn.4.
5. Equate Eqn.4 to Eqn.1.
6. Manipulate algebraically to
show that T2 = constant x R3
My 3rd law
has me
baffled!
Then try a dose
of NLUG and
see your
problems vanish!
Problem solving with NLUG (Take G = 6.7 x 10-11 N m2 kg-2)
Q1. (a) Calculate the force of gravitational attraction that you
exert on the person next to you. (1 stone = 6.4 kg)
Q1. (b) How big is the force of gravitational attraction that s/he exerts
on you?
Q2. (a) What force does the Earth exert on a 1 kg mass on its
surface?
Q2. (b) What force does Mars exert on this 1 kg mass?
Data:
Earth mass = 6 x 1024 kg; Mars mass = 6.4 x 1023 kg;
Closest Earth-Mars approach distance = 7.8 x 1010 m
Q3. In light of your answers to Q2, should you be worried by a
planetary alignment?
Gravitational fields
• Explain that a gravitational
field is a region in space
where a mass feels a force
due to another mass
• Describe and explain the
variation of the radial field
with distance from a massive
body
• Use field considerations in
calculations of satellite orbit
radii, asteroid masses etc.
Geostationary orbits
(Take G = 6.7 x 10-11 N m2 kg-2)
A satellite placed in a geostationary orbit remains over the same point
on the equator as the Earth rotates.
Q1. Use the equation you derived for Kepler’s 3rd law to calculate the correct orbit
radius and height above the Earth’s surface for such a satellite.
Q2. Calculate the strength of the Earth’s gravitational field at this location.
Q3. Using a diagram or otherwise, calculate the minimum number of geostationary
satellites needed for complete coverage of the Earth.
Data: Mass of Earth = 6 x 1024 kg Radius of Earth = 6360 km
Determining asteroid masses (Take G = 6.7 x 10-11 N m2 kg-2)
A probe is in orbit 300 km above the
centre of a spherical asteroid. It takes
314 minutes to make one orbit around
the asteroid.
Q1. Show that the orbital speed of the
probe is about 100 ms-1.
Q2. Hence show that the centripetal
acceleration of the probe is about 0.033
ms-2.
Q3. Explain why the gravitational field
strength due to the asteroid at the
probe’s orbit radius must be 0.033 Nkg-1.
Q4. Use your answer to Q3 to show that
the asteroid’s mass is about 4.4 x 1019 kg.
Why does the gun
recoil?
..and which member of
the crew needs to study
Chapter 11?
What will happen when the moving
spacecraft docks with the
stationary one?
Momentum
• Recap principle of
momentum
conservation
• Apply to a solve
problems for a range
of situations where
bodies interact
BEFORE
10kg
AFTER
2 ms-1
1 ms-1 10 kg
5 kg not moving
5 kg
6 ms-1
Q1. Is momentum conserved in this collision?
Q2. Can you nevertheless explain why this collision violates a
fundamental law of physics, and therefore would never take
place?
Q1. Predict what will happen when a light object travelling to
the right strikes a stationary object that is 10 times more massive.
(You can explore this with physics-online/virtual air track)
Q2. Suppose the mass of the heavy ball is increased until it is 100
times more massive than the light ball. What happens now?
(You can explore this with Modellus/Activity 160S)
Q3. Explain the relevance of the
above to “gravitational sling
shotting” of spacecraft by a planet.
Impulse
Impulse is a measure of the change of momentum of a
body when a resultant force acts for a certain time.
Starting with Newton’s 2nd law (F = ma) and a relevant
suvat equation, derive an expression for the change
in momentum of a body in terms of
the force acting on it and the time for
which the force acts.
Impulse
• Show that impulse = change
in momentum = F x Δt
• Review Newton’s 3rd law
and its applications
• Solve problems involving
impulse considerations,
including ball sports and
rocket motion examples
How does FΔt = mΔv help us
understand the following...
Force-time graph for tennis racket
hitting a ball
(N)
(s)
How can we calculate change in
momentum from the graph?
How does FΔt = mΔv help us
understand the following...
Note that when two objects
collide/interact, the change in
momentum of one body is equal in
size but opposite in direction to the
change in momentum of the other
Impulse questions
Q1. A ball of mass 0.06 kg moving at 15 ms-1 hits a wall at right angles and bounces off along the
same line at 10 ms-1.
(a) What is the magnitude of the impulse of the wall on the ball?
(b) What is the magnitude of the impulse of the ball on the wall?
(c) If the ball is in contact with the wall for 3 x 10-2 seconds, estimate the average force on the ball.
(d) Sketch a graph showing how the force on the ball would vary with time during the impact.
(e) Is it possible to calculate the velocity with which the wall “moves away” after the impact? What
information would you need?
Q2. The ion engine on a spaceship in deep space produces an accelerating force of 20 N for 100
seconds.
(a) Calculate the impulse provided by the ion engine.
(b) If the spaceship has a mass of 10 000 kg and is initially travelling at 2000 ms-1, what is its final
velocity after the 100 second thrust phase?
(c) What do you conclude about the usefulness of ion drives in spacecraft propulsion?
Gravitational potential
• Explain the meaning of the
term, and how it is calculated
for radial fields
• Describe the relationship
between field strength and
potential gradient
• Use kinetic-potential energy
exchange considerations to
explain escape velocity,
cometary orbits, energetics of
satellite orbits etc.