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UNIFORM CIRCULAR MOTION
THE MOTION OF AN OBJECT
moving IN A CIRCLE AT
CONSTANT SPEED
Average speed= distance/time=arc/time =
=*R/time=2R/T(period)
where R is the radius of the circumference
and T is called the period of the circular motion
Related formulas
Angular Speed Formula calculates the distance traveled by the body in
terms of rotations to the time taken. It is represented by ω and is given as
angular speed= 𝜔 =/t
Distance traveled is in terms of angle θ is measured in radians and time
taken is in seconds. Hence the Angular speed is expressed in radians per
seconds or rad/s.
Angular speed for one complete rotation is given as 2𝜋/T
The relation between Linear speed and Angular speed is v=𝜔𝑅
Where v = Linear speed and
r = radius of circular path.
Does constant speed mean constant velocity?
• Speed and velocity refer to two distinctly different quantities. Speed
is a scalar quantity and velocity is a vector quantity. Velocity, being a
vector, has both a magnitude and a direction. The magnitude of the
velocity vector is the instantaneous speed of the object. The
direction of the velocity vector is directed in the same direction that
the object moves. Since an object is moving in a circle, its direction
is continuously changing. At one moment, the object is moving
northward such that the velocity vector is directed northward. One
quarter of a cycle later, the object would be moving eastward such
that the velocity vector is directed eastward. As the
object rounds the circle, the direction of the velocity vector is
different than it was the instant before. So while the magnitude of
the velocity vector may be constant, the direction of the velocity
vector is changing. The best word that can be used to describe the
direction of the velocity vector is the word tangential.
SPEED and VELOCITY
• To summarize, an object moving in uniform
circular motion is moving around the
perimeter of the circle with a constant speed.
While the speed of the object is constant, its
velocity is changing. Velocity, being a vector,
has a constant magnitude but a changing
direction. The direction is always directed
tangent to the circle and as the object turns
the circle, the tangent line is always pointing
in a new direction
ACCELERATION?
• Sure!
• the fact is that an accelerating object is an object
that is changing its velocity. And since velocity is a
vector that has both magnitude and direction, a
change in either the magnitude or the direction
constitutes a change in the velocity. For this
reason, it can be safely stated that an object
moving in a circle at constant speed is indeed
accelerating. It is accelerating because the
direction of the velocity vector is changing.
An object moving in a circle is experiencing an acceleration. Even if
moving around the perimeter of the circle with a constant speed,
there is still a change in velocity and subsequently an acceleration.
This acceleration is directed towards the center of the circle. And in
accord with Newton's second law of motion, an object which
experiences an acceleration must also be experiencing a net force.
The direction of the net force is in the same direction as the
acceleration. So for an object moving in a circle, there must be an
inward force acting upon it in order to cause its inward
acceleration. This is sometimes referred to as the centripetal force
requirement. The word centripetal (not to be confused with the Fwordcentrifugal) means center seeking. For object's moving in
circular motion, there is a net force acting towards the center
which causes the object to seek the center.
Mathematics of Circular Motion
• There are three mathematical quantities that
will be of primary interest to us as we analyze
the motion of objects in circles. These three
quantities are speed, acceleration and force.
• Average speed
v=2R/T
• Acceleration=𝑣 2 /R
• Force=m𝑣 2 /R