A person standing at the top of a cliff whirls a stone at the end of a string in a horizontal circle. The stone is released at a point 20.0 m above the base of the cliff and lands a horizontal distance X from the base. X is thirty times the radius of the circle on which the stone is whirled.
Determine the angular speed of the stone at the moment of release.
To determine the angular speed of the stone at the moment of release, we need to use the concept of conservation of energy.
Let's break down the problem step by step:
1. Determine the potential energy at the top of the cliff:
The potential energy at the top of the cliff is given by the formula: PE = mgh, where m is the mass of the stone, g is the acceleration due to gravity, and h is the height of the cliff. In this case, the stone is released, so its potential energy is converted entirely into kinetic energy.
2. Determine the kinetic energy of the stone just before release:
The kinetic energy of the stone just before release is given by the formula: KE = (1/2)mv^2, where m is the mass of the stone and v is the velocity of the stone. Since the stone moves horizontally in a circle, the velocity v is tangential to the circle.
3. Equate the potential energy to kinetic energy:
Setting the potential energy equal to the kinetic energy, we have mgh = (1/2)mv^2. The mass cancels out, and we are left with gh = (1/2)v^2.
4. Relate the velocity to the angular speed:
The velocity v is related to the angular speed ω (omega) by the formula: v = rω, where r is the radius of the circle on which the stone is whirled. In this case, X is thirty times the radius, so we have v = 30rω.
5. Substitute the velocity equation into the energy equation:
Substituting v = 30rω into gh = (1/2)v^2, we have gh = (1/2)(30rω)^2. Simplifying, we get gh = (450r^2)ω^2.
6. Solve for the angular speed ω:
Rearranging the equation, we have ω^2 = (gh) / (450r^2). Taking the square root of both sides, we get ω = √(gh / 450r^2).
Now, we have the equation for the angular speed of the stone at the moment of release in terms of the given variables: ω = √(gh / 450r^2).
You can substitute the values of g, h, and r into this equation to find the numerical value of the angular speed.