Respuesta :
Assuming that we have a circular path (rotational motion), the angular velocity (w) can be calculated as follows:
w = v/r
From this relation, we can see that the angular velocity is inversely proportional to the radius. This means that if the radius increases, the angular velocity decreases and vice versa.
Based on this, if the radius is doubles, the angular velocity would decrease to half its value, assuming that v is constant.
w = v/r
From this relation, we can see that the angular velocity is inversely proportional to the radius. This means that if the radius increases, the angular velocity decreases and vice versa.
Based on this, if the radius is doubles, the angular velocity would decrease to half its value, assuming that v is constant.
The angular speed at twice the initial radius becomes half of the initial angular speed.
Further Explanation:
Speed is the measure of a quantity of an object the tells how fast the object is moving in the other words we can define the speed that it is the distance covered by an body divided by the time taken to cover that distance. It is a quantity with only magnitude so it is a scalar quantity.
Given:
The certain angular speed is [tex]{\omega _2}[/tex].
The radius is [tex]r[/tex].
Concept:
The expression for the linear motion can be written as:
[tex]\fbox{\begin\\v=\dfrac{s}{t}\end{minispace}}[/tex] …… (1)
Here, [tex]v[/tex] is the linear speed, [tex]s[/tex] is the total distance covered and [tex]t[/tex] is the time taken to cover to distance.
The expression for the circular speed can be written as:
[tex]\fbox{\begin\\\omega=\dfrac{\theta }{t}\end{minispace}}[/tex]
Here, [tex]\omega[/tex] is the circular speed and [tex]\theta[/tex] is the angular displacement.
The expression for the total distance covered in term of angular displacement is:
[tex]s=\theta r[/tex]
Substitute [tex]\theta r[/tex] for [tex]s[/tex] in equation (1)
[tex]\begin{aligned}v&=\frac{{\theta r}}{t}\hfill\\v&=\frac{\theta }{t}\cdot r\hfill\\v&=\omega\cdot r\hfill\\\omega&=\frac{v}{r} \hfill\\ \end{aligned}[/tex]
From above expression the angular speed is inversely proportional to the radius.
[tex]\fbox{\begin\\\omega\propto\dfrac{1}{r}\end{minispace}}[/tex]
That is if the radius increases the angular speed decreases and if the radius decreases the angular speed increases.
Considered the linear speed remain content.
Case 1:
The angular speed is given by
[tex]{\omega _1}=\dfrac{v}{r}[/tex] …… (2)
Case 2:
The radius is double that is [tex]2r[/tex].
The angular speed is given by
[tex]{\omega _2}=\dfrac{v}{{2r}}[/tex] …… (3)
Divide equation (2) by equation (3).
[tex]\begin{aligned}\frac{{{\omega _2}}}{{{\omega _1}}}&=\frac{{\frac{v}{{2r}}}}{{\frac{v}{r}}}\hfill\\\frac{{{\omega _2}}}{{{\omega _1}}}&=\frac{1}{2}\hfill\\{\omega _2}&=\frac{{{\omega _1}}}{2}\hfill\\\end{aligned}[/tex]
Therefore, the angular speed at twice the initial radius becomes half of the initial angular speed.
Learn more:
1. Angular speed https://brainly.in/question/7744338
2. Angular velocity https://brainly.com/question/11209367
3. Angular speed https://brainly.com/question/4721004
Answer Details:
Grade: college
Subject: Physics
Chapter: Kinematics
Keywords:
Certain, angular speed, w2, radius, r, twice, w1/2, half, initial angular speed.
