Resource Lesson
Centripetal Acceleration and Angular Motion
Printer Friendly Version
Suppose instead of looking at the rotation of an entire merry-go-round, we examine only the change in the position of one rider on its surface, shown by the orange dot. For this initial discussion, we are going to assume that the merry-go-round is rotating at a constant rate so that the rider moves through a circular path, or linear circumference, at a constant speed. Please be conscious of the fact that the rider's velocity is not constant since the direction of her motion is constantly changing as shown in the second diagram.
Image courtesy of
What ever happened to the merry-to-round?
As long as the rider does not move along the merry-go-round's surface, the magnitude of the rider's velocity (that is, her speed) equals
where
is the merry-go-round's constant angular velocity and
r
is the rider's radial distance from the center of the merry-go-round.
Although the merry-go-round has no angular acceleration, the rider is experiencing a
centripetal acceleration
towards the center of the circle, or the axis of rotation. This type of acceleration acts parallel to the radius of the circle and is often referred to as
radial acceleration
.
As with linear acceleration, centripetal acceleration also points in the direction of the change in velocity. Vectorially, this is confirmed in the diagram to the left where you see the vector
pointing to the center of the circle. Remember that
and that subtracting a vector just means that you add the same magnitude vector pointing in 180º the opposite direction.
This type of acceleration is called
uniform centripetal acceleration
since the object's speed is not changing, just its direction is changing at a uniform rate based on the merry-go-round's angular velocity.
If the force causing this centripetal (or radial - "along the radius") acceleration were to be removed, the object would "fly off at a tangent" and no longer move in a circular path.
As shown in the previous lesson, an object's centripetal acceleration is calculated with the formula
. Since
we can rewrite this equation as
.
If asked to calculate the rider's
total linear acceleration
, you would also need to examine any tangential acceleration present,
. In our case, the angular acceleration equals zero since we are demanding that the merry-go-round rotate at a constant angular velocity. Since
and the rider's net acceleration just equals the center-seeking acceleration along the radius, or
.
Now let's examine a rotational system which is experiencing angular acceleration. We know from our previous study of rotational dynamics, that an angular acceleration requires a torque. Our model will be a fixed pulley that has a string wrapped around its perimeter from which a mass is suspended. When the mass is released, a torque will be provided by the tension in the string resulting in a counterclockwise rotation of the pulley.
A freebody diagram of the falling mass would show two forces: T and mg. Since the mass is falling, mg must be greater than the tension, T.
Solving for tension we have
Since the string does not slip over the pulley, but causes it to rotate, we know that
.
In this example, a "piece of dust" remaining in place on the rim of the pulley would experience two types of linear acceleration: centripetal acceleration directed towards the center of the pulley and tangential acceleration parallel to the edge of the pulley.
Since these two accelerations are at right angles to each other, the
net linear acceleration
is calculated using the Pythagorean Theorem.
Also note that this value would be an instantaneous value since the centripetal acceleration is dependent on the instantaneous angular/linear velocity.
Related Documents
Lab:
Labs -
A Physical Pendulum, The Parallel Axis Theorem and A Bit of Calculus
Labs -
Calculation of "g" Using Two Types of Pendulums
Labs -
Conical Pendulums
Labs -
Conical Pendulums
Labs -
Conservation of Energy and Vertical Circles
Labs -
Conservation of Momentum in Two-Dimensions
Labs -
Density of an Unknown Fluid
Labs -
Introductory Simple Pendulums
Labs -
Kepler's 1st and 2nd Laws
Labs -
Loop-the-Loop
Labs -
Mass of a Paper Clip
Labs -
Moment of Inertia of a Bicycle Wheel
Labs -
Oscillating Springs
Labs -
Roller Coaster, Projectile Motion, and Energy
Labs -
Rotational Inertia
Labs -
Sand Springs
Labs -
Simple Pendulums: Class Data
Labs -
Simple Pendulums: LabPro Data
Labs -
Video LAB: A Gravitron
Labs -
Video LAB: Circular Motion
Labs -
Video LAB: Looping Rollercoaster
Labs -
Water Springs
Resource Lesson:
RL -
A Chart of Common Moments of Inertia
RL -
A Derivation of the Formulas for Centripetal Acceleration
RL -
A Further Look at Angular Momentum
RL -
Center of Mass
RL -
Conservation of Energy and Springs
RL -
Derivation of Bohr's Model for the Hydrogen Spectrum
RL -
Derivation: Period of a Simple Pendulum
RL -
Discrete Masses: Center of Mass and Moment of Inertia
RL -
Energy Conservation in Simple Pendulums
RL -
Gravitational Energy Wells
RL -
Hinged Board
RL -
Introduction to Angular Momentum
RL -
Kepler's Laws
RL -
LC Circuit
RL -
Magnetic Forces on Particles (Part II)
RL -
Period of a Pendulum
RL -
Rolling and Slipping
RL -
Rotary Motion
RL -
Rotational Dynamics: Pivoting Rods
RL -
Rotational Dynamics: Pulleys
RL -
Rotational Dynamics: Rolling Spheres/Cylinders
RL -
Rotational Equilibrium
RL -
Rotational Kinematics
RL -
Rotational Kinetic Energy
RL -
SHM Equations
RL -
Simple Harmonic Motion
RL -
Springs and Blocks
RL -
Symmetries in Physics
RL -
Tension Cases: Four Special Situations
RL -
The Law of Universal Gravitation
RL -
Thin Rods: Center of Mass
RL -
Thin Rods: Moment of Inertia
RL -
Torque: An Introduction
RL -
Uniform Circular Motion: Centripetal Forces
RL -
Universal Gravitation and Satellites
RL -
Vertical Circles and Non-Uniform Circular Motion
Review:
REV -
Review: Circular Motion and Universal Gravitation
Worksheet:
APP -
Big Al
APP -
Ring Around the Collar
APP -
The Baton Twirler
APP -
The Satellite
APP -
The See-Saw Scene
APP -
The Spring Phling
APP -
Timex
CP -
Center of Gravity
CP -
Centripetal Acceleration
CP -
Centripetal Force
CP -
Satellites: Circular and Elliptical
CP -
Torque Beams
CP -
Torque: Cams and Spools
NT -
Center of Gravity
NT -
Center of Gravity vs Torque
NT -
Circular Orbits
NT -
Falling Sticks
NT -
Pendulum
NT -
Rolling Cans
NT -
Rolling Spool
NT -
Rotating Disk
NT -
Spiral Tube
WS -
Basic Practice with Springs
WS -
Inertial Mass Lab Review Questions
WS -
Introduction to Springs
WS -
Kepler's Laws: Worksheet #1
WS -
Kepler's Laws: Worksheet #2
WS -
Moment Arms
WS -
Moments of Inertia and Angular Momentum
WS -
More Practice with SHM Equations
WS -
Pendulum Lab Review
WS -
Pendulum Lab Review
WS -
Practice: SHM Equations
WS -
Practice: Uniform Circular Motion
WS -
Practice: Vertical Circular Motion
WS -
Rotational Kinetic Energy
WS -
SHM Properties
WS -
Static Springs: The Basics
WS -
Torque: Rotational Equilibrium Problems
WS -
Universal Gravitation and Satellites
WS -
Vertical Circular Motion #1
TB -
Basic Torque Problems
TB -
Center of Mass (Discrete Collections)
TB -
Centripetal Acceleration
TB -
Centripetal Force
TB -
Moment of Inertia (Discrete Collections)
TB -
Rotational Kinematics
TB -
Rotational Kinematics #2
PhysicsLAB
Copyright © 1997-2018
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton