UNIFORM CIRCULAR MOTION OF A MATERIAL POINT
Circular motion is motion along a curved path in the shape of a circle, in contrast to rectilinear motion where the body moves along a straight path. In this article, an animation of the circular motion of a material point will be shown. Unlike rectilinear motion, where the physical quantities describing the motion are distance s [m], velocity v [m/s], and acceleration a [m/s²], in circular motion we consider quantities such as the angle φ [rad] between the position vector and the X-axis, angular velocity ω [rad/s], and angular acceleration α [rad/s²].
The simplest form of curved motion of a material point is circular motion. At every point on the path, the direction of the instantaneous velocity matches the tangent to the path. The velocity vector and the radius vector are perpendicular to each other.
Types of circular motion:
- uniform circular motion – the magnitude of the velocity does not change
- non-uniform circular motion – the magnitude of the velocity changes
If a material point moves along a circular path with constant speed, such motion is called uniform circular motion.
Since in uniform circular motion the magnitude of velocity is constant, the tangential acceleration is zero. However, the direction of velocity continuously changes, so there is a normal (centripetal) acceleration.
The direction of normal acceleration coincides with the radius of the circle, and it always points toward the center of the circle. That is why it is called radial or centripetal acceleration.
The magnitude of centripetal acceleration is directly proportional to the square of the velocity of the material point and inversely proportional to the radius of the path along which the point moves.
ac = v2 / r
UNIFORMLY ACCELERATED CIRCULAR MOTION OF A MATERIAL POINT
If a material point moves along a circle with a velocity whose magnitude changes uniformly, such motion is called uniformly accelerated circular motion.
Since in uniformly accelerated circular motion the magnitude of velocity changes, there is both normal (radial or centripetal) and tangential acceleration. The magnitude of tangential acceleration is constant, while the magnitude of normal acceleration is not (see formula – it varies with speed).
aT = Δv / Δt
aT = (v2 - v1) / (t2 - t1) = const
Resultant acceleration vector: \( \vec{a} = \vec{a}_n + \vec{a}_t \)
a = √ac2 + aT2
a = √(v2/r)2 + aT2
In uniformly accelerated circular motion:
- the direction of tangential acceleration and velocity coincide
- the resultant acceleration vector and the velocity vector form an acute angle
Angular velocity is a physical quantity that describes the rate of rotation of a material point.
Angular acceleration is a physical quantity that describes the change of angular velocity in variable rotational motion.
CENTRIPETAL FORCE
The direction of normal acceleration coincides with the radius and points toward the center of the circle. Since there is acceleration, there must also be a force causing it. This force has the same direction and orientation as the normal acceleration. Because it is directed toward the center of rotation, it is called the centripetal force.
Magnitude of centripetal force:
Fcp = m * v2 / r
Another formula for centripetal force:
Fcp = m * r * ω2
The string exerts a centripetal force on the ball, forcing it to move in a circle.
If the string breaks, the ball will fly off tangentially in a straight line at the velocity it had at that moment.If there is also a tangential force, it will cause the body to accelerate (tangential acceleration will appear).
Tangential force changes the magnitude of the speed of the body moving along the circular path, while the centripetal force changes the direction of the tangential velocity.
Centripetal force is not a specific type of force but rather a name for any force whose effect causes circular motion of a body around some center. Centripetal force is the force that forces a body to move along a circular path.