A rock of mass m = 1.8 kg is tied to a string of length L = 2.4 m and is twirled in a vertical circle as shown in the figure below. The speed v of the rock is constant; that is, it is the same at the top and the bottom of the circle. If the tension in the string is zero when the rock is at its highest point (so that the string just barely goes slack), what is the tension when the rock is at the bottom?
2 M g
V^2/R must equal M*g in order for string tension to go to zero at the top. At the bottom, the same centripetal force gets added to the weight (M*g) when computing string tension.
To find the tension when the rock is at the bottom, let's consider the forces acting on the rock at that point.
1. First, we need to determine the forces acting on the rock at the bottom of the circle. These forces include gravity (mg) and tension in the string (T).
2. At the bottom of the circle, the rock is moving in a circular path. In order to maintain this motion, there must be a net inward force acting towards the center of the circle. This force is the tension in the string (T).
3. The force of gravity acts vertically downward and can be divided into two components: one along the string and another perpendicular to it. The component along the string does not affect the tension, while the perpendicular component (mg * cosθ) acts in the opposite direction of the tension.
4. The net inward force can be found by subtracting the perpendicular component of gravity (mg * cosθ) from the tension force (T).
5. At the highest point of the circle, the tension is zero, so we can set the net inward force to zero as well.
6. Setting the net inward force to zero, we have:
T - mg * cosθ = 0
7. Now, we need to find the value of cosθ at the bottom of the circle. Since the rock is twirled in a vertical circle, at the bottom, the tension force is opposite to the gravitational force. Therefore, cosθ = -1.
8. Substituting cosθ = -1 into our equation, we have:
T - (-mg) = 0
T + mg = 0
9. Solving for T, we get:
T = -mg
10. The negative sign indicates that the tension force is downward, opposing the gravitational force.
Therefore, the tension when the rock is at the bottom is equal to the magnitude of the gravitational force, which is T = mg. In this case, the tension is T = (mass of the rock) * (acceleration due to gravity), that is, T = (1.8 kg) * (9.8 m/s^2) = 17.64 N.