PHYA11H3 Chapter Notes - Chapter 8: Circular Motion, Angular Acceleration, Angular Velocity
12 Dec 2016
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Chapter 8 Dynamics II: Motion in a Plane
8.1 Dynamics in Two Dimensions
- The x- and y-components of the acceleration vector are given by:
And,
o Suppose the x- and y-components of acceleration are independent of each other, therefore ax
does not depend on either y or vy, and similarly ay does not depend on x or vx
Problem Solving Strategy
1. Draw a pictorial representation –a sketch and a free-body diagram
2. Use Newton’s second law in component form:
(Fnet) = =max and (Fnet) = =may
3. Solve for the acceleration. If the acceleration is constant, use the two-dimensional kinematic
equations to find velocities and positions
Projectile Motion
- Vertical motion is free fall, while the horizontal motion is one of constant velocity
o However, the situation is quite different for a low-mass projectile, where the effects of drag
are too large to ignore
o The acceleration of a projectile subject to drag is:
Here the components of acceleration are not independent of each other because ax depends on vy
and vice versa
8.2 Uniform Circular Motion
- Recall that a particle in uniform circular motion with angular velocity ω has speed v= ωr and
centripetal acceleration
- Let’s establish a coordinate system with its origin at the point where the particle is located
o The axis are defined as:
▪ The r-axis (radial axis) points from the particle toward the center of the circle
▪ The t-axis (tangential axis) is tangent to the circle, pointing in the ccw direction
▪ The z-axis is perpendicular to the plane of motion
The three axes of this rtz-coordinate system are mutually perpendicular, just like the axes
of the familiar xyz-coordinate system
- Notice how the axes move with the particle so that the r-axis always points to the center of the circle
- Recall, a particle in uniform circular motion has a velocity tangential to the circle and an
acceleration –the centripetal acceleration –pointing toward the center of the circle
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o Thus the rtz-components of and are
Where ω=dθ/dt, the angular velocity, must be in rad/s
▪ In other words, the velocity vector has only a tangential component, the acceleration
vetor has only a radial component
▪ ar is referred to as the centripetal acceleration
Dynamics of Uniform Circular Motion
- Newton’s second law tells us how much net force is required to cause centripetal acceleration:
net=m=
In other words, a particle of mass m moving at constant speed v around a circle of radius r must
have a net force of magnitude mv2/r pointing toward the center of the circle –without such a force, the
particle would move off in a straight line tangent to the circle
- Note: A particle moves with uniform circular motion if and only if a net force always points
towards the center of the circle
o The force itself must have an identifiable agent and will be one of our familiar forces, such
as tension, friction, or the normal force
- For uniform circular motion, the sum of the forces along the t-axis and along the z-axis must equal
zero, and the sum of the forces along the r-axis must equal mar, where ar is the centripetal
acceleration
8.3 Circular Orbits
- Orbit: a closed trajectory around a planet or a star
o An orbiting projectile is in free fall
- In the flat-earth approximation, the gravitational force acting on an object of mass m is
G = (mg, vertically downward)
But since stars and planets are actually spherical, the real force of gravity acting on an object is
directed toward the center of the planet
G = (mg, toward center)
find more resources at oneclass.com
find more resources at oneclass.com
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Document Summary
Chapter 8 dynamics ii: motion in a plane. If the acceleration is constant, use the two-dimensional kinematic. Here the components of acceleration are not independent of each other because ax depends on vy. Recall that a particle in uniform circular motion with angular velocity has speed v= r and. The three axes of this rtz-coordinate system are mutually perpendicular, just like the axes of the familiar xyz-coordinate system. Notice how the axes move with the particle so that the r-axis always points to the center of the circle. Recall, a particle in uniform circular motion has a velocity tangential to the circle and an acceleration the centripetal acceleration pointing toward the center of the circle: thus the rtz-components of (cid:1874) and are. Where =d /dt, the angular velocity, must be in rad/s. Net=m =(cid:4672)(cid:3040)2 ,(cid:1867)(cid:1875)(cid:1856) (cid:1855)(cid:1857)(cid:1866)(cid:1857) (cid:1867)(cid:1858) (cid:1855)(cid:1855)(cid:1864)(cid:1857)(cid:4673: ar is referred to as the centripetal acceleration.
