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.
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To determine the angular speed of the stone at the moment of release, we can use the conservation of energy.
Let's break down the problem step by step:
1. First, we need to find the speed of the stone when it reaches the base of the cliff. Since the stone is released at a point 20.0 m above the base, it falls freely under the influence of gravity. The potential energy lost by the stone is given by mgh, where m is the mass of the stone, g is the acceleration due to gravity (9.8 m/s^2), and h is the height (20.0 m).
2. The potential energy lost by the stone is converted into kinetic energy, given by (1/2)mv^2, where v is the speed of the stone when it reaches the base.
3. Since the stone is moving in a horizontal circle, the centripetal force required is provided by tension in the string. At the moment of release, the tension is the only force acting on the stone horizontally. The centripetal force required for the circular motion is mv^2/r, where r is the radius of the circle.
4. Equating the centripetal force to the tension, we have mv^2/r = Tension.
5. We can solve for velocity (v) in terms of tension and radius (r). Substituting for Tension using T = mg + Tension, where T is the total tension in the string and mg is the force due to gravity, we get v^2 = (T - mg)r/m.
6. The radius of the circle is given as X/30.
7. Since the stone is moving in a horizontal circle, the centripetal acceleration is related to the angular speed (ω) by a = rω^2.
8. Substituting for the centripetal acceleration using a = g, we get rω^2 = g.
9. Solving for ω (angular speed), we get ω = √(g/r).
Substituting the value of r = X/30 into the equation for ω, we can calculate the angular speed of the stone at the moment of release.