a stone tied at one end of a string of length,l is whirled in a horizontal circle with constant linear speed,v.which of the following expressions describe the relation between the radius of the circle,r and the angle between the string and the vertical axis?
To understand the relation between the radius of the circle, r, and the angle between the string and the vertical axis, let's break down the situation.
When the stone is being whirled in a horizontal circle, it experiences two forces: the tension force from the string and the gravitational force pulling it downward.
The tension force acts as the centripetal force, which keeps the stone moving in a circular path. The tension force is always directed towards the center of the circle. On the other hand, the gravitational force acts vertically downward towards the ground.
Since these forces are perpendicular to each other, we can analyze their components separately. The tension force can be split into two components: one acting horizontally (towards the center) and another acting vertically (opposing gravity).
- The horizontal component of the tension force provides the centripetal force: Fc = MASS * (V^2 / R), where M is the mass of the stone, V is the linear speed, and R is the radius of the circle.
- The vertical component of the tension force balances out the gravitational force: Fg = MASS * g, where g is the acceleration due to gravity.
Now, let's consider the angle between the string and the vertical axis. Let's call this angle "θ." We can break the tension force into its vertical and horizontal components using trigonometry.
The vertical component of the tension force can be calculated as: Tsin(θ).
The horizontal component of the tension force (centripetal force) can be calculated as: Tcos(θ).
Equating these forces with the formulas mentioned above:
Tsin(θ) = MASS * g
Tcos(θ) = MASS * (V^2 / R)
Now, if we solve for T in the first equation and substitute it into the second equation, we get:
(MASS * g / sin(θ)) * cos(θ) = MASS * (V^2 / R)
Simplifying:
g * cos(θ) / sin(θ) = V^2 / R
tan(θ) = (V^2 / (g * R))
Therefore, the relation between the radius of the circle, R, and the angle between the string and the vertical axis, θ, is:
tan(θ) = (V^2 / (g * R))
In summary, the expression that describes the relation between the radius of the circle, R, and the angle between the string and the vertical axis, θ, is tan(θ) = (V^2 / (g * R)).