# Relating Linear and Angular Kinematics

• Fernando Rios
In summary: So, if ##v## is constant, what is ##a##?In summary, we use the definitions of linear and angular speed, linear acceleration, and average angular acceleration to solve for the angular speed, length of track, and average angular acceleration of a compact disc (CD) spinning at a constant linear speed of 1.25 m/s. The angular speed is 50 rad/s when scanning the innermost part of the track and 21.55 rad/s when scanning the outermost part of the track. The length of the track on a maximum-duration CD would be 5.55 km. The average angular acceleration of a maximum-duration CD during its 74.0-min playing time is 0.00641 rad/s
Fernando Rios
Homework Statement
A compact disc (CD) stores music in a coded pattern of tiny pits m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. a) What is the angular speed of theCDwhen scanning the innermost part of the track? The outermost part of the track? b) The maximum playing time of a CD is 74.0 min. What would be the length of the track on such a maximum-duration CD if it were
stretched out in a straight line? c) What is the average angular acceleration of a maximum-duration CD during its 74.0-min playing time? Take the direction of rotation of the disc to be positive.
Relevant Equations
v = r*omega
v = d/t
alpha = (omega_f - omega_0)/t
a) We use the definition of linear speed in terms of angular speed:
v = r*omega

omega_f = v/r = (1.25 m/s)/(0.025 m) = 50 rad/s

omega_0 = v/r = (1.25 m/s)/(0.025 m) = 21.55 rad/s

b) We use the definition of linear speed:
v = d/t

d = vt = (1.25m/s)(74 min)(60 s/1 min) = 5.55 km

c) We use the definition of average angular acceleration:
alpha = (omega_f - omega_0)/t = (50 rad/s- 21.55 rad/s)/(74 min)(1 min/60 s) = 0.00641 rad/s^2

The answers are correct. I just wonder, why if a = r*alpha and alpha has a numerical value, a is still equal to zero (there is constant linear velocity)?

Fernando Rios said:
Homework Statement:: A compact disc (CD) stores music in a coded pattern of tiny pits m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. a) What is the angular speed of theCDwhen scanning the innermost part of the track? The outermost part of the track? b) The maximum playing time of a CD is 74.0 min. What would be the length of the track on such a maximum-duration CD if it were
stretched out in a straight line? c) What is the average angular acceleration of a maximum-duration CD during its 74.0-min playing time? Take the direction of rotation of the disc to be positive.
Relevant Equations:: v = r*omega
v = d/t
alpha = (omega_f - omega_0)/t

why if a = r*alpha
This is only true if r is constant. In your case, r is not constant.

As v must be kept constant for proper reading, and v = r*omega, the angular velocity (omega or amount of angle swept each second) and the radius must keep an inverse proportion.

As r increases, the machine needs to slow the rotation down, in order to keep the condition of constant velocity of reading.
For the same reason, as r decreases, the machine needs to speed the rotation up.

http://hyperphysics.phy-astr.gsu.edu/hbase/rotq.html#avel

:)

Fernando Rios said:
The answers are correct. I just wonder, why if a = r*alpha and alpha has a numerical value, a is still equal to zero (there is constant linear velocity)?
Think about the case of uniform circular motion. Do you have constant linear velocity? Note that for uniform circular motion ##a=v^2/r##.

## 1. What is the difference between linear and angular kinematics?

Linear kinematics deals with the motion of objects in a straight line, while angular kinematics deals with the motion of objects in a circular or rotational path.

## 2. How are linear and angular kinematics related?

Linear and angular kinematics are related through the concept of velocity. In linear kinematics, velocity is measured in meters per second, while in angular kinematics, it is measured in radians per second. However, both types of velocity represent the rate of change of position over time.

## 3. What is the formula for converting linear velocity to angular velocity?

The formula for converting linear velocity (v) to angular velocity (ω) is ω = v/r, where r is the radius of the circular path.

## 4. How does acceleration differ in linear and angular kinematics?

In linear kinematics, acceleration is the rate of change of velocity, while in angular kinematics, it is the rate of change of angular velocity. Additionally, linear acceleration is measured in meters per second squared, while angular acceleration is measured in radians per second squared.

## 5. How can we use linear and angular kinematics in real-life applications?

Linear and angular kinematics are used in various real-life applications, such as in sports to analyze the motion of athletes, in engineering to design and analyze machines and structures, and in robotics to program the movement of robotic arms. They are also used in navigation systems, such as GPS, to calculate the position and speed of moving objects.

Replies
23
Views
507
Replies
3
Views
1K
Replies
7
Views
1K
Replies
3
Views
674
Replies
3
Views
1K
Replies
2
Views
5K
Replies
10
Views
1K
Replies
4
Views
2K
Replies
4
Views
1K
Replies
18
Views
5K