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A centripetal force (from Latin centrum, "center" and petere, "to seek" [1]) is a force that makes a body follow a curved path.The direction of the centripetal force is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path.
In physics, angular acceleration (symbol α, alpha) is the time rate of change of angular velocity.Following the two types of angular velocity, spin angular velocity and orbital angular velocity, the respective types of angular acceleration are: spin angular acceleration, involving a rigid body about an axis of rotation intersecting the body's centroid; and orbital angular acceleration ...
This acceleration is known as centripetal acceleration. For a path of radius r , when an angle θ is swept out, the distance traveled on the periphery of the orbit is s = rθ . Therefore, the speed of travel around the orbit is v = r d θ d t = r ω , {\displaystyle v=r{\frac {d\theta }{dt}}=r\omega ,} where the angular rate of rotation is ω .
For a given angular velocity, the centripetal acceleration is directly proportional to radius . This is due to the dependence of velocity v {\displaystyle v} on the radius r {\displaystyle r} . v = ω r . {\displaystyle v=\omega r.}
Transverse acceleration (perpendicular to velocity) causes a change in direction. If it is constant in magnitude and changing in direction with the velocity, circular motion ensues. Taking two derivatives of the particle's coordinates concerning time gives the centripetal acceleration = =
Timing diagram over one revolution for angle, angular velocity, angular acceleration, and angular jerk. Consider a rigid body rotating about a fixed axis in an inertial reference frame. If its angular position as a function of time is θ(t), the angular velocity, acceleration, and jerk can be expressed as follows:
When considered in an inertial frame (that is to say, one that is not rotating with the Earth), the non-zero acceleration means that force of gravity will not balance with the force from the spring. In order to have a net centripetal force, the magnitude of the restoring force of the spring must be less than the magnitude of force of gravity.
The trivial case of the angular momentum of a body in an orbit is given by = where is the mass of the orbiting object, is the orbit's frequency and is the orbit's radius.. The angular momentum of a uniform rigid sphere rotating around its axis, instead, is given by = where is the sphere's mass, is the frequency of rotation and is the sphere's radius.