a crane holds a 500kg wrecking ball in place using a wire with tension T1 and is horizontal to the ground. The crane swings it with a cable under tension T2. if the wire holds the ball in place so the cable forms a 20 degree angle with the vertical, determine tensions T1 and T2.
To determine the tensions T1 and T2, we can apply the principles of equilibrium. In this situation, we have a crane holding a wrecking ball using a wire and swinging it with a cable.
Let's break down the forces acting on the wrecking ball:
1. Weight (mg): The weight of the wrecking ball is acting vertically downward. The magnitude can be calculated using the formula: weight = mass * gravity. Given that the mass of the ball is 500 kg and assuming gravity is 9.8 m/s^2, the weight is 500 kg * 9.8 m/s^2 = 4900 N.
2. Tension T1: The wire is holding the ball in place horizontally. Since the ball is at rest in that position, the horizontal component of T1 must balance the weight of the ball. Therefore, T1 = weight = 4900 N.
3. Tension T2: The cable is swinging the ball, forming an angle of 20 degrees with the vertical direction. We need to consider two components of tension T2: horizontal and vertical.
a. Horizontal component (T2x): This component acts in the same direction as the horizontal component of T1, balancing it. Therefore, T2x = T1 = 4900 N.
b. Vertical component (T2y): This component counteracts the weight of the ball. To find T2y, we can use trigonometry. Since the angle between T2 and the vertical is 20 degrees, T2y = T2 * sin(20 degrees).
Now, let's calculate T2y using the calculated value of T2x:
T2y = T2 * sin(20 degrees) = T2x
Since T2x = T1 = 4900 N, we can rewrite the equation as:
T2 * sin(20 degrees) = 4900 N
Now, we can solve for T2:
T2 = 4900 N / sin(20 degrees)
Using a calculator:
T2 ≈ 14359.7 N
Therefore, the tension T2 in the cable is approximately 14359.7 N, and the tension T1 in the wire holding the ball horizontally is 4900 N.