A bowling ball weighing 71.3 is attached to the ceiling by a rope of length 3.77 . The ball is pulled to one side and released; it then swings back and forth as a pendulum. As the rope swings through the vertical, the speed of the bowling ball is 5.00 .
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To find the tension in the rope, we can use the concept of conservation of energy. At the highest point of its swing (when the rope is horizontal), all of the potential energy is converted into kinetic energy.
The potential energy at the highest point is given by the equation:
Potential energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)
In this case, the height is the length of the rope, 3.77 meters, and the mass is 71.3 kg. The acceleration due to gravity is approximately 9.8 m/s^2. Therefore, the potential energy at the highest point is:
PE = 71.3 kg * 9.8 m/s^2 * 3.77 m = 2622.36 J (Joules)
At the lowest point of its swing (when the rope is vertical), all of the potential energy is converted into kinetic energy. The kinetic energy is given by the equation:
Kinetic energy (KE) = 0.5 * mass (m) * velocity^2 (v^2)
In this case, the velocity is given as 5.00 m/s. Therefore, the kinetic energy at the lowest point is:
KE = 0.5 * 71.3 kg * (5.00 m/s)^2 = 887.5 J
Since the energy is conserved, the potential energy at the highest point is equal to the kinetic energy at the lowest point. Therefore, we can set up the equation:
PE = KE
2622.36 J = 0.5 * 71.3 kg * (5.00 m/s)^2
Solving for the missing value, we get:
Tension in the rope = 2622.36 J / (0.5 * 71.3 kg * (5.00 m/s)^2)
Tension in the rope ≈ 291.82 N
Therefore, the tension in the rope is approximately 291.82 Newtons.