PHYSICS 1A
Popmintchev
Lecture 5B
Newtons Laws. Friction.
REVIEW OF LAST LECTURE
N  normal force always points perpendicular to
surface
Ff  friction force always points parallel to surface
T  tension always points along string
Break 3D problem into
PHYSICS 1A
Popmintchev
Lecture 2B
1D > 2D Motion
OUTLINE
LAST LECTURE
v = dx/dt
a = dv/dt = d2x/dt2
Velocity = slope of x(t)
Acceleration = slope of v(t)
Position>Velocity>Acceleration related by d/dt
Acceleration>Velocity>Position related by
PHYSICS 1A
Popmintchev
Lecture 4A
Kinematics > Dynamics
REVIEW OF LAST LECTURE
Uniform circular motion: v=const, acceleration
Radial acceleration change in direction of v
Tangential acceleration  change in the
magnitude of v
Radial acceleration = c
PHYSICS 1A
Popmintchev
Lecture 3B
2D Rotational Motion
OUTLINE
LAST LECTURE
Motion projected on different axes
independent kinematic equations
Prodigious choice of coordinate system
Time of flight. Range.
TODAY
Intersecting trajectories
Circular m
PHYSICS 1A
Popmintchev
Lecture 1A
Physical Quantities
Information
Todays Lecture
Overview
Physical quantities
Units
Dimensional analysis
Motion diagrams
Tutorial center: MonThu 1 5 pm
Physics: Discipline that attempts to explain
a wide range of natural
Physics 1A
Spring 2017: Course Policy
1
UNIVERSITY OF CALIFORNIA SAN DIEGO
DEPARTMENT OF PHYSICS
Physics 1A
CLASSICAL MECHANICS
INSTRUCTOR:
Prof. Tenio Popmintchev
[email protected]
Office: Mayer Hall 3641
Office Hours: Friday 1:00 pm  3
PHYSICS 1A
Popmintchev
Lecture 3A
2D Motion
OUTLINE
LAST LECTURE
Velocity and acceleration in 1D
Kinematic equations
1D > 2D
TODAY
Choice of coordinate systems
2D motion (ballistic, time of flight)
Use results from one direction to use in second
PHYSICS 1A
Popmintchev
Lecture 4B
Newtons Laws
WebAssign Home Page
https:/youtu.be/mj18qOZVO7g
OVERVIEW
LAST LECTURE
Newtons first two simple laws
Objects move with constant velocity (magnitude
and direction) unless acted upon by a force
Acceleration
PHYSICS 1A
Popmintchev
Lecture 1B
Vectors
Outline
Last Lecture
Physical quantities
Fundamental quantities of length, time and mass
Dimensional analysis is a useful trick for
guessing a solution or checking your answer
Motion diagrams
This Lecture
S
Vector Functions and Curves
Applications
Question
What are the tangential and normal components of the Coriolis force on an
object moving with (horizontal) velocity v in the following situations
(a) At the north pole.
(b) At the south pole.
(c) At the equ
Vector Functions and Curves
Applications
Question
If an object moves with position vector r(t) satisfying
dr
= a (r(t) b)
dt
and with r(0) = r 0 .
a, b and r0 are constant vectors with a 6= 0.
Describe the path along which this object moves.
Answer
dr
d

Vector Functions and Curves
Applications
Question
If run at full power, a selfpropelled tank car can accelerate itself, when full
(mass = M kg) along a straight track at a m/sec2 .
At time t = 0 the tank is full. The contents leave the tank at a rate k k
Vector Functions and Curves
Applications
Question
Solve the initial value problem
dr
= kr
dt
r(0) = i + k
Describe the curve r = r(t).
Answer
dr
= kr
dt
r(0) = i + k
If r(t) = x(t)i + y(t)j + z(t)k
Then
x(0) = z(0) = 1
y(0) = 0
As k(dr/dt) = k(kr) = 0, ve
Vector Functions and Curves
Applications
Question
A satellite in a low (radius of orbit is approximately the radius of the Earth)
circular orbit passes over both poles. It takes the satellite two hours to make
one revolution.
If an observer stands on the
Vector Functions and Curves
One variable functions
Question
A particle is moving around a circle at constant speed. Given that the equation of the circle is x2 + y 2 = 25 and that the particle makes one revolution
every two seconds, find its acceleration
Vector Functions and Curves
One variable functions
Question
If the position and velocity of a particle satisfy r v > 0, what does this tell
you about the motion of the particle. What if r v < 0?
Answer
If r v > 0 then r is increasing.
So r is moving awa
Vector Functions and Curves
One variable functions
Question
Find the velocity, speed and acceleration of the particle with position given
by r(t) at time t. Also determine the particles path.
r = t2 i t2 j + k
Answer
Position: r = t2 i t2 j + k
Velocity:
Vector Functions and Curves
One variable functions
Question
A particle moves along the curve y = 3/x, travelling to the right. At the
point (2, 23 ) its speed is 10, what is its velocity?
Answer
When its xcoordinate is x, the object is at position
r = xi
Vector Functions and Curves
One variable functions
Question
An object travels on the curve of intersection of the cylinders y = x2 and
z = x2 with increasing x. When the particle is at (1, 1, 1), it has a speed of
9cm/s which is increasing at a rate of 3c
Vector Functions and Curves
One variable functions
Question
Find the velocity, speed and acceleration of the particle with position given
by r(t) at time t. Also determine the particles path.
r = t2 j + tk
Answer
Position: r = t2 j + tk
Velocity: v =
2tj
Vector Functions and Curves
One variable functions
Question
Find the velocity, speed and acceleration of the particle with position given
by r(t) at time t. Also determine the particles path.
r = i + tj + tk
Answer
Position: r = i + tj + tk
Velocity: v =
Vector Functions and Curves
One variable functions
Question
A particle travels along the curve of intersection of the plane x + y = 2 and
the cylinder z = x2 in the direction of increasing y. The particle has constant
speed v = 3, what is its velocity at
Vector Functions and Curves
One variable functions
Question
Find the velocity, speed and acceleration of the particle with position given
by r(t) at time t. Also determine the particles path.
r = aet i + bet j + cet k
Answer
Position: r = aet i + bet j +
Vector Functions and Curves
One variable functions
Question
Show that if the scalar product of the velocity and acceleration of an object
in motion is negative (or positive) then the speed of the object is decreasing
(or increasing).
Answer
d
d 2
v = v
Vector Functions and Curves
One variable functions
Question
It is given that the position and velocity vectors of a moving object satisfy
v(t) = 2r(t) for all times t. If r(0) = r 0 , find r(t) and a(t), the acceleration.
Also determine the path of motion
Vector Functions and Curves
One variable functions
Question
A particle is travelling along the curve y = x2 , z = x3 and has constant
vertical speed w = dz/dt = 3. When the particle is at the point (2, 4, 8), find
both its velocity and acceleration.
Answe
Vector Functions and Curves
One variable functions
Question
Given that the position and velocity vectors of a moving object are always
perpendicular, show that the objects path lies on a sphere.
Answer
d
d 2
r = r r = 2r v = 0
dt
dt
r is constant.
He
Vector Functions and Curves
One variable functions
Question
Find the velocity, speed and acceleration of the particle with position given
by r(t) at time t. Also determine the particles path.
r = at cos ti + at sin tj + b ln 4k
Answer
Position: r = at cos
Vector Functions and Curves
One variable functions
Question
Find the velocity, speed and acceleration of the particle with position given
by r(t) at time t. Also determine the particles path.
r = a cos ti + bj + a sin tk
Answer
Position: r = a cos ti + bj