Course Outline :
•Analysis Model: Particle in Uniform Circular Motion
•Tangential and Radial Acceleration
•Extending the Particle in Uniform Circular Model
Chapter 6 Circular Motion and Other
Applications’ of Newton’s Laws
6.1 Uniform circular motion
Uniform circular motion occurs when an object moves in a
circular path with a constant speed
An acceleration exists since the direction of the motion is
changing
This change in velocity is related to an acceleration
The velocity vector is always tangent to the path of the
object
Changing Velocity in Uniform Circular
Motion
The change in the velocity
vector is due to the
change in direction
The vector diagram
shows Dv = vf - vi
Centripetal Acceleration
The acceleration is always perpendicular to the path of the
motion
The acceleration always points toward the center of the circle
of motion
This acceleration is called the centripetal acceleration
Centripetal Acceleration, cont
The magnitude of the centripetal acceleration vector is given
by
The direction of the centripetal acceleration vector is always
changing, to stay directed toward the center of the circle of
motion
2
C
va
r
Period
The period, T, is the time required for one complete
revolution
The speed of the particle would be the circumference of the
circle of motion divided by the period
Therefore, the period is
2 rT
v
4.5 Tangential Acceleration
The magnitude of the velocity could also be changing
In this case, there would be a tangential acceleration
Total Acceleration The tangential
acceleration causes the
change in the speed of
the particle
The radial acceleration
comes from a change in
the direction of the
velocity vector
Total Acceleration, equations
The tangential acceleration:
The radial acceleration:
The total acceleration:
Magnitude
t
da
dt
v
2
r C
va a
r
2 2
r ta a a
Total Acceleration, In Terms of Unit
Vectors
Define the following unit
vectors
r lies along the radius vector
q is tangent to the circle
The total acceleration is
ˆˆ andr q
2
ˆ ˆt r
d v
dt rq v
a a a r
Analysis Model: Particle in Uniform Circular Motion
Based on Newton’s 2nd Law, there must have net force acting
on the particle to cause the acceleration. Other wise, the
particle will remain at rest or move with constant velocity.
Conical Pendulum
The object is in equilibrium in the vertical direction .
It undergoes uniform circular motion in the horizontal direction.
∑Fy = 0 →T cos θ = mg
∑Fx = T sin θ = m ac
v is independent of m
Section 6.1
Example (6.1) : The Conical Pendulum
A small object of mass m is suspended from a string of length L. The object revolves with constant
speed v in a horizontal circle of radius r (Figure (6.3)). (Because the string sweeps out the surface of
a cone, the system is known as a conical pendulum). Find an expression for v.
Example (6.2) : How Fast Can it Spin?
A ball of mass 0.500 kg is attached to the end of a cord 1.50 m long. The ball is whirled in a horizontal
circle as was shown in Figure (6.1). If the cord can withstand a maximum tension of 50.0N, what is
the maximum speed the ball can attain before the cord breaks? Assume that the string remains
horizontal during the motion.
Example 1: A car of mass m round a curve on a flat, horizontal road of
radius R. If the coefficient of static friction between tires and the road is μs,
what is the maximum speed vmax at which the car can take the curve without
sliding?
The force that enables the car to move in curve is the static friction force
because no slipping occurs at the point of contact between road and tires. If
the car were on an icy road, it would only move in straight line
6.2 Non-Uniform Circular Motion
The acceleration and force have tangential components.
produces the centripetal acceleration
produces the tangential acceleration
The total force is
Section 6.2
Vertical Circle with Non-Uniform Speed
The gravitational force exerts a tangential force on the object. Look at the components of Fg
Model the sphere as a particle under a net force and moving in a circular path. Not uniform circular motion
The tension at any point can be found.
Section 6.2
Example 3: A particle of mass m attached to the end of a cord of length l whirls in a
vertical circle. Find the tension in the cord and the tangential acceleration when the
speed of the particle is v and the cord makes an angle θ with the vertical.
Note : The tension, friction, normal, gravitational, electric, and
magnetic forces are examples of possible centripetal forces that
enable an object to move in circular path.