Welcome to Physics I !!!

1
Physics I95.141LECTURE 1010/12/10
2
Exam Prep Problem (Conical Pendulum)

• A mass (m2kg) suspended on a cord (l1m)
  revolves in a circle of radius r. (?30º)
• A) (10pts) Draw a free body diagram for the mass,
  labeling your coordinate system
• B) (5pts) What is the Tension in the cord?
• C) (5pts) What is the acceleration of the ball,
  and in what direction?
• D) (5pts) What is the tangential velocity of the
  ball?

3
Exam Prep Problem (Conical Pendulum)

• A mass (m2kg) suspended on a cord (l1m)
  revolves in a circle of radius r. (?30º)
• A) (10pts) Draw a free body diagram for the mass,
  labeling your coordinate system

4
Exam Prep Problem (Conical Pendulum)

• A mass (m2kg) suspended on a cord (l1m)
  revolves in a circle of radius r. (?30º)
• B) (5pts) What is the tension in the cord?

5
Exam Prep Problem (Conical Pendulum)

• A small mass (m) suspended on a cord (l1m)
  revolves in a circle of radius r. (?30º)
• C) (5pts) What is the acceleration of the ball,
  and in what direction?

6
Exam Prep Problem (Conical Pendulum)

• A small mass (m) suspended on a cord (l1m)
  revolves in a circle of radius r. (?30º)
• D) (5pts) What is the tangential velocity of the
  ball?

7
Outline

• Highway curves (circular motion)
• Newtons Law of Universal Gravitation
• Weightlessness
• Keplers Laws
• What do we know?
• Units
• Kinematic equations
• Freely falling objects
• Vectors
• Kinematics Vectors Vector Kinematics
• Relative motion
• Projectile motion
• Uniform circular motion
• Newtons Laws
• Force of Gravity/Normal Force
• Free Body Diagrams
• Problem solving
• Uniform Circular Motion

8
Highway Curves

• In order for the car to make a curve without
  slipping/skidding, need sufficient Force from
  friction.
• This force is a static friction, even though the
  car is moving!!
• Coordinate system!!!

9
Flat Curves (unbanked)

• What is the coefficient of static friction
  required to make an unbanked curve with radius R,
  for a car traveling with a speed v?

10
Banked Curves

• Can a car make a turn on a banked frictionless
  surface without skidding? For speed v, radius R,
  what angle is required?
• Coordinate system!!

11
Example Problem

• A car goes around an unbanked curve (R100m) at a
  speed of 50m/s. The concrete/tire interface has
  a coefficient of static friction of 1. Can the
  car make this turn?

12
Example Problem

• A car goes around an banked curve (R100m) at a
  speed of 50m/s. Ignoring friction, what angle
  should the curve be banked at to allow the car to
  make the curve?

13
Newtons Law of Universal Gravitation

• We know that falling objects accelerate.
• We also know that if an object accelerates, there
  must be a force acting on it.
• The Force that accelerates falling bodies is
  gravity.
• But what exerts this force?
• Since all falling objects fall towards the center
  of the Earth, Newton suggested that it is the
  Earth itself which is exerting this Force.

14
Newtons Law of Universal Gravitation

• What form does this Force take?
• 1) Dependence on distance
• Newton knew that the moon orbited the Earth.
• We know that a circular motion requires an inward
  radial acceleration

15
Newtons Law of Universal Gravitation

• So if the Force causing the moon to orbit the
  Earth, is the same force which causes object to
  accelerate at the surface of the Earth, then this
  Force goes as the inverse square of the distance
  from the center of the Earth

16
Newtons Law of Universal Gravitation

• The other thing we should note, is that the Force
  due to gravity produces the same acceleration for
  ALL OBJECTS, regardless of mass. So this is a
  Force which must scale with the mass of the
  object.

• Symmetry (Newtons 3rd Law) also suggests that
  this Force must depend on the mass of the Earth,
  or the second body.

17
Newtons Law of Universal Gravitation

• Finally, Newton argues that if this is the Force
  causing the moon to orbit the Earth, perhaps it
  is also the Force causing the planets to Orbit
  the sun. In fact, perhaps every mass exerts a
  gravitational force on every other mass in the
  universe.
• And we can write this force as

18
Direction of Gravitational Force

• Force is a vector, and therefore has a magnitude
  and direction.
• Direction is along line connecting two masses.

19
Example Problem

• Imagine 3 Blocks, of equal mass, placed at three
  corners of a square. Draw the gravitational
  Force vectors acting on each block.

1
3
2
20
Gravitational Attraction Between Two People

• Tom Cruise (160lbs) and Katie Holmes (118lbs) are
  dancing (about 0.5 m apart). What is attractive
  Force of Gravity between them?

21
Gravity at Earths Surface

• If this law is correct, what should we get for Fg
  at the Earths surface?

22
Satellites

• Imagine I throw a ball with some horizontal
  velocity vo.
• In previous chapters, we studied projectile
  motion, which tells us the ball will accelerate
  towards earth and eventually fall to Earth.
• But this is an approximationthe Earth is not
  flat, and the Force of gravity is not downward,
  but towards the center of the Earth.

23
Satellites

• In order for an object to travel with uniform
  circular motion, a radial Force is required.
• What speed would the ball need to have to travel
  in a circular path at the surface of the Earth?

24
Satellites

• The uniform circular motion due to gravitation is
  known as an orbit, and an orbiting object is
  often referred to as a satellite.
• We can calculate the speed of a satellite for a
  given orbital radius

25
Geosynchronous Orbit

• A geosynchronous orbit, is one that orbits at the
  same speed the Earth rotates, so that the
  satellite stays at the same position with respect
  to Earth as it orbits. How can we calculate the
  height of a satellite of mass M in geosynchronous
  orbit?

26
Weightlessness

• Remember, we define weight as the magnitude of
  the Force of gravity acting on an object.
• At the surface of Earth this is mg.
• But we measure weight, by measuring the Force a
  mass exerts on a scale.

• Imagine we are weighing a mass in an elevator.
• If the elevator is at rest, or moving at a
  constant velocity, what does the scale read?

27
Weightlessness

• Now, if the elevator is accelerating upwards with
  ag/2.
• What is the mass apparent weight?

28
Weightlessness

• Now, if the elevator is in freefall (a-g).
• What is the mass apparent weight?

29
Weightlessness

• The weightlessness one experiences in orbit is
  exactly the same one would feel in a freely
  falling elevator.
• Remember, the Force causing a satellite to orbit
  is the Force of Gravity.in essence, the
  satellite is freely falling.

30
Keplers Laws

• Keplers laws of planetary motion
• Empirical (Experimental)
• Keplers 1st Law The path of each planet around
  the sun is an ellipse with the sun at one focus

a ? semi-major axis b ? semi-minor axis e ?
eccentricity e ? es/a
es
a
b
31
Keplers Laws

• Keplers 2nd Law Each planet moves so that an
  imaginary line drawn from the sun to the planet
  sweeps out equal areas in equal periods of time

32
Keplers Laws

• Keplers 3rd Law The ratio of the squares of the
  periods of any two planets revolving around the
  sun is equal to the ratio of the cubes of their
  semi-major axes.

s2
s1
33
Newtons Synthesis

• Newton was able to derive Keplers experimental
  laws from his Universal Law of Gravity.
• Perturbations

34
Keplers 3rd Law Circular Orbit

• Earths orbit around sun has e0.017, almost
  circular. Can we prove Keplers third law for
  circular orbits?

35
Example Problem

• If we know the Earths distance from the sun, can
  we determine the suns mass from Keplers laws?