When a rigid body rotates about a fixed axis, every particle in the body moves in a circular path. As shown in the figure below, the arclength As between the angular positions 0, and 0, is given as As = rA8. Do you agree with this equation? rotational motion (a) In the limit where Að is very small, then As can be consider as a straight line. Is this true or false? (b) It follows from statement (a) that any points along the circular path there is a tangential velocity (v) that is always perpendicular to the radius of the rotating body. Hence, if we divide both sides of the equation As = rÃo by At, justify that we will get vy = rw. This result indicates that the direction of the particle's velocity is tangential to its circular path at each point. Most importantly, for example, this result tells us the relation- ship between the angular velocity of the wheel of the car and linear velocity of the car. Reference line - A0 At (e) If the angular velocity changes by A, then the rotating object's linear speed will change by Arg. Hence, we will have An = rAw. Is this a true statement? (d) If this changes takes place in some smallAt and if we divide both sides of the equation Ar = rAw by At, justify that we will get ae = ra, where az is called the tangential acceleration and a is the rigid body angular acceleration. Most importantly, for example, this result tells us the relationship between the angular acceleration of the wheel of the car and the linear acceleration of the car.
When a rigid body rotates about a fixed axis, every particle in the body moves in a circular path. As shown in the figure below, the arclength As between the angular positions 0, and 0, is given as As = rA8. Do you agree with this equation? rotational motion (a) In the limit where Að is very small, then As can be consider as a straight line. Is this true or false? (b) It follows from statement (a) that any points along the circular path there is a tangential velocity (v) that is always perpendicular to the radius of the rotating body. Hence, if we divide both sides of the equation As = rÃo by At, justify that we will get vy = rw. This result indicates that the direction of the particle's velocity is tangential to its circular path at each point. Most importantly, for example, this result tells us the relation- ship between the angular velocity of the wheel of the car and linear velocity of the car. Reference line - A0 At (e) If the angular velocity changes by A, then the rotating object's linear speed will change by Arg. Hence, we will have An = rAw. Is this a true statement? (d) If this changes takes place in some smallAt and if we divide both sides of the equation Ar = rAw by At, justify that we will get ae = ra, where az is called the tangential acceleration and a is the rigid body angular acceleration. Most importantly, for example, this result tells us the relationship between the angular acceleration of the wheel of the car and the linear acceleration of the car.
College Physics
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Author:Paul Peter Urone, Roger Hinrichs
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Chapter6: Uniform Circular Motion And Gravitation
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Problem 16PE: Olympic ice skaters are able to spin at about 5 rev/s. (a) What is their angular velocity in radians...
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The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
Question
Transcribed Image Text:7. When a rigid body rotates about a fixed axis, every particle in the body moves in a circular path.
As shown in the figure below, the arclength As between the angular positions 6, and 82 is given as
As = rA8. Do you agree with this equation?
rotational motion
(a) In the limit where A6 is very small, then As can be
consider as a straight line. Is this true or false?
AsrAe
(b) It follows from statement (a) that any points along the
circular path there is a tangential velocity (vr) that is
always perpendicular to the radius of the rotating body.
Hence, if we divide both sides of the equation As = rAe
by At, justify that we will get v = rw. This result indicates
that the direction of the particle's velocity is tangential
to its circular path at each point. Most importantly, for example, this result tells us the relation-
ship between the angular velocity of the wheel of the car and linear velocity of the car.
Reference line
At
(c) If the angular velocity changes by Aw, then the rotating object's linear speed will change by Avr.
Hence, we will have Aer = rAw. Is this a true statement?
(d) If this changes takes place in some small At and if we divide both sides of the equation Av = rAw
by At, justify that we will get at = ra, where a is called the tangential acceleration and a is the
rigid body angular acceleration. Most importantly, for example, this result tells us the relationship
between the angular acceleration of the wheel of the car and the linear acceleration of the car.
(e) When an object is under a rotational motion or circular motion, that object besides contains
tangential acceleration it also has centripetal acceleration as well. In the figure below, for example,
the child on the merry-go-round has both tangential and centripetal acceleration. The equation
for the centripetal acceleration is ap = v7/r (we are not going to derive this equation), and its
direction of acceleration is toward the center. Using Newton's second law, this gives the centripetal
force as Fep = map or Fep = . Does this make sense? What is the direction of the centripetal
force on the child?
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