Circular Motion
8.01
Position and Displacement
r
r (t ) : position vector
o f a n o b je c t m o v in g in a
circular orbit of radius R
r
!r (t ): change in position between time t and
time t+t
P o s i ti o n v e c to r i s c h a n g i n g i n d i r e c t
Circular Motion
Concept Questions
Question 1 A bead is given a small push at the top of a hoop (position A) and is
constrained to slide around a frictionless circular wire (in a vertical plane). Circle the
arrow that best describes the direction of the ac
Circular Motion
Concept Questions
Question 1 A bead is given a small push at the top of a hoop (position A) and is
constrained to slide around a frictionless circular wire (in a vertical plane). Circle the
arrow that best describes the direction of the ac
Problem Solving Circular Motion Kinematics
Challenge Problem Solutions
Problem 1
A bead is given a small push at the top of a hoop (position A) and is constrained to slide
around a frictionless circular wire (in a vertical plane). Circle the arrow that be
Problem Solving Circular Motion Kinematics
Challenge Problems
Problem 1
A bead is given a small push at the top of a hoop (position A) and is constrained to slide
around a frictionless circular wire (in a vertical plane). Circle the arrow that best
descri
Chapter 9 Uniform Circular Motion
9.1 Introduction
Special cases often dominate our study of physics, and circular motion is certainly no
exception. We see circular motion in many instances in the world; a bicycle rider on a
circular track, a ball spun ar
Circular Motion Dynamics
Strategy: Circular Orbits
i) Understand geometry
ii) From geometry determine
acceleration
iii) Find combination of forces that give
acceleration
2
Strategy:
Applying Newtons Second Law for
Circular Motion
Always has a compon
Circular Motion Dynamics
Concept Questions
Problem 1: A puck of mass m is moving in a circle at constant speed on a frictionless
table as shown above. The puck is connected by a string to a suspended bob, also of mass
m , which is at rest below the table.
Circular Motion Dynamics
Concept Questions
Problem 1: A puck of mass m is moving in a circle at constant speed on a frictionless
table as shown above. The puck is connected by a string to a suspended bob, also of mass
m , which is at rest below the table.
Problem Solving Circular Motion Dynamics
Problem 1: Double Star System
Consider a double star system under the influence of gravitational force between the
stars. Star 1 has mass m1 and star 2 has mass m2 . Assume that each star undergoes
uniform circular
Problem Solving Circular Motion Dynamics
Challenge Problems
Problem 1: Double Star System
Consider a double star system under the influence of gravitational force between the
stars. Star 1 has mass m1 and star 2 has mass m2 . Assume that each star undergo
Module 10: Circular Motion Dynamics
10.1 Newtons Second Law and Circular Motion
We have already shown that when an object moves in a circular orbit of radius R with
angular velocity ! , it is most convenient to choose polar coordinates to describe the
pos
Central Force Motion
Central Force Problem
Find the motion of two bodies interacting via a central force.
Examples:
Gravitational force (Kepler problem):
r
m1m2
F1,2 (r ) = !G 2 r
r
Linear restoring force:
r
F1,2 (r ) = !kr r
Two Body Problem: Center
of M
Kepler Problem
Concept Questions
Question1 Which of the following are Keplers Laws?
A. Each planet moves in an elliptical orbit, with the sun at the center of the ellipse.
B. Each planet moves in an elliptical orbit, with the sun at the focus of the ell
Kepler Problem
Concept Questions
Question1 Which of the following are Keplers Laws?
A. Each planet moves in an elliptical orbit, with the sun at the center of the ellipse.
B. Each planet moves in an elliptical orbit, with the sun at the focus of the ell
Central Force Motion
Challenge Problems
Problem 1: Elliptic Orbit
A satellite of mass ms is in an elliptical orbit around a planet of mass m p which is
located at one focus of the ellipse. The satellite has a velocity va at the distance ra when
it is furt
Central Force Motion
Challenge Problems
Problem 1: Elliptic Orbit
A satellite of mass ms is in an elliptical orbit around a planet of mass m p which is
located at one focus of the ellipse. The satellite has a velocity va at the distance ra when
it is furt
Module 28: The Kepler Problem: Planetary Mechanics
28.1 Introduction Keplers Laws:1
1. Each planet moves in an ellipse with the sun at one focus.
2. The radius vector from the sun to a planet sweeps out equal areas in equal time.
3. The period of revol
Cartesian Coordinate System
and Vectors
Coordinate System
Coordinate system: used to describe the position
of a point in space and consists of
1. An origin as the
reference point
2. A set of coordinate
axes with scales and
labels
3. Choice of positive
Vectors
Concept Questions
Question 1. Consider the pair of units vectors ( P , P ) located at the point P , and the
ij
pair of units vectors ( , ) located at the point S . Which of the following statements is
ij
S
S
true?
3)
!
iP iS
!
jP jS
=
i
i
4)
=
Vectors
Concept Questions
Question 1. Consider the pair of units vectors ( P , P ) located at the point P , and the
ij
pair of units vectors ( , ) located at the point S . Which of the following statements is
ij
S
true?
3)
!
iP iS
!
j P jS
=
i
i
4)
=
Problem Solving Vectors
Challenge Problem Solutions
Problem 1: Vector Addition
!
1.1 Consider the two vectors shown in the figure below. The magnitude of A = 2.88
!
and the vector A makes an angle 33.7! with the positive x -axis. The magnitude of
!
!
B =
Problem Solving Vectors
Challenge Problems
Problem 1: Vector Addition
!
1.1 Consider the two vectors shown in the figure below. The magnitude of A = 2.88 and the
!
!
vector A makes an angle 33.7! with the positive x -axis. The magnitude of B = 3.44 and t
Module 3: Cartesian Coordinates and Vectors
Philosophy is written in this grand book, the universe which stands
continually open to our gaze. But the book cannot be understood unless
one first learns to comprehend the language and read the letters in whic
Application of Newtons
Second Law
Newtons Second Law
The change of motion is proportional to the
motive force impresses, and is made in the direction
of the right line in which that force is impressed,
r
r
F = m a.
When multiple forces are acting,
r
r
Fi
Application of Newtons Laws
Concept Questions
Question 1 Two !
blocks 1 and 2, on a frictionless table, are pushed from the left by a
!
horizontal force F1 , and on the right by a horizontal force of magnitude F2 as shown
!
!
above. The magnitudes of the
Application of Newtons Laws
Concept Questions
Question 1 Two !
blocks 1 and 2, on a frictionless table, are pushed from the left by a
!
horizontal force F1 , and on the right by a horizontal force of magnitude F2 as shown
!
!
above. The magnitudes of the
Application of Newtons Second Law
Challenge Problem Solutions
Problem 1: Painter on a Platform
A painter of mass m1 stands on a platform of mass m2 and pulls himself up by two ropes
that run over massless pulleys, as shown. He pulls on each rope with a fo
Application of Newtons Second Law
Challenge Problems
Problem 1: Painter on a Platform
A painter of mass m1 stands on a platform of mass m2 and pulls himself up by two ropes
that run over massless pulleys, as shown. He pulls on each rope with a force of ma
Module 8: Newtons Laws of Motion
8.1 Newtons First Law
The First Law of Motion, commonly called the Principle of Inertia, was first realized
by Galileo. (Newton did not acknowledge Galileos contribution.) Newton was
particularly concerned with how to phra