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- University physics volume 1
- Unit 1. mechanics
- Fixed-axis rotation
- Rotational variables
We now have a basic vocabulary for discussing fixed-axis rotational kinematics and relationships between rotational variables. We discuss more definitions and connections in the next section.
Summary
- The angular position
of a rotating body is the angle the body has rotated through in a fixed coordinate system, which serves as a frame of reference.
- The angular velocity of a rotating body about a fixed axis is defined as
, the rotational rate of the body in radians per second. The instantaneous angular velocity of a rotating body
is the derivative with respect to time of the angular position
, found by taking the limit
in the average angular velocity
. The angular velocity relates
to the tangential speed of a point on the rotating body through the relation
, where
r is the radius to the point and
is the tangential speed at the given point.
- The angular velocity
is found using the right-hand rule. If the fingers curl in the direction of rotation about a fixed axis, the thumb points in the direction of
(see
[link] ).
- If the system’s angular velocity is not constant, then the system has an angular acceleration. The average angular acceleration over a given time interval is the change in angular velocity over this time interval,
. The instantaneous angular acceleration is the time derivative of angular velocity,
. The angular acceleration
is found by locating the angular velocity. If a rotation rate of a rotating body is decreasing, the angular acceleration is in the opposite direction to
. If the rotation rate is increasing, the angular acceleration is in the same direction as
.
- The tangential acceleration of a point at a radius from the axis of rotation is the angular acceleration times the radius to the point.
Conceptual questions
A clock is mounted on the wall. As you look at it, what is the direction of the angular velocity vector of the second hand?
The second hand rotates clockwise, so by the right-hand rule, the angular velocity vector is into the wall.
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A baseball bat is swung. Do all points on the bat have the same angular velocity? The same tangential speed?
They have the same angular velocity. Points further out on the bat have greater tangential speeds.
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The blades of a blender on a counter are rotating clockwise as you look into it from the top. If the blender is put to a greater speed what direction is the angular acceleration of the blades?
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Problems
A wheel rotates at a constant rate of
. (a) What is its angular velocity in radians per second? (b) Through what angle does it turn in 10 s? Express the solution in radians and degrees.
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A particle moves 3.0 m along a circle of radius 1.5 m. (a) Through what angle does it rotate? (b) If the particle makes this trip in 1.0 s at a constant speed, what is its angular velocity? (c) What is its acceleration?
a.
; b.
; c.
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A compact disc rotates at 500 rev/min. If the diameter of the disc is 120 mm, (a) what is the tangential speed of a point at the edge of the disc? (b) At a point halfway to the center of the disc?
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Unreasonable results. The propeller of an aircraft is spinning at 10 rev/s when the pilot shuts off the engine. The propeller reduces its angular velocity at a constant
for a time period of 40 s. What is the rotation rate of the propeller in 40 s? Is this a reasonable situation?
The propeller takes only
to come to rest, when the propeller is at 0 rad/s, it would start rotating in the opposite direction. This would be impossible due to the magnitude of forces involved in getting the propeller to stop and start rotating in the opposite direction.
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On takeoff, the propellers on a UAV (unmanned aerial vehicle) increase their angular velocity from rest at a rate of
for 3.0 s. (a) What is the instantaneous angular velocity of the propellers at
? (b) What is the angular acceleration?
a.
; b.
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The angular position of a rod varies as
radians from time
. The rod has two beads on it as shown in the following figure, one at 10 cm from the rotation axis and the other at 20 cm from the rotation axis. (a) What is the instantaneous angular velocity of the rod at
(b) What is the angular acceleration of the rod? (c) What are the tangential speeds of the beads at
(d) What are the tangential accelerations of the beads at
(e) What are the centripetal accelerations of the beads at
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Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
what are the types of wave
Maurice
fine, how about you?
Mohammed
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
Who can show me the full solution in this problem?
Source:
OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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