A 2520 kg demolition ball swings at the end of a 23.9 m cable on the arc of a vertical circle. At the lowest point of the swing, the ball is moving at a speed of 8.97 m/s. Determine the tension in the cable.
Enough, try some yourself and post your attempts if you get stuck please.
To determine the tension in the cable, we can start by analyzing the forces acting on the demolition ball at the lowest point of the swing.
At the bottom of the swing, the ball is moving in a circular path, so it experiences centripetal acceleration towards the center of the circle. The centripetal force required to maintain this motion is provided by the tension in the cable.
The gravitational force also acts on the ball, directed downwards. At the lowest point, the tension and gravitational force combine to provide the net force required for the circular motion.
To find the tension in the cable, we can use the equation for the net force acting on an object:
Net Force = Tension - Weight
where Weight = mass × gravitational acceleration.
In this case, the mass of the demolition ball is given as 2520 kg, and the gravitational acceleration is approximately 9.8 m/s².
We also know that at the lowest point, the speed of the ball is given as 8.97 m/s. This speed can be related to the radius of the circular path using the centripetal acceleration formula:
Centripetal acceleration = (velocity²) / radius
We can rearrange this equation to solve for the radius:
radius = (velocity²) / Centripetal acceleration
Using this equation, we can calculate the radius of the circular path.
Once we have the radius, we can substitute the values into the Net Force equation and solve for the tension.
Let's do the calculations step by step:
Step 1: Calculate the radius of the circular path:
radius = (velocity²) / Centripetal acceleration
= (8.97 m/s)² / (9.8 m/s²)
= 8.13 m (approximately)
Step 2: Calculate the weight:
Weight = mass × gravitational acceleration
= 2520 kg × 9.8 m/s²
= 24756 N
Step 3: Calculate the tension in the cable:
Net Force = Tension - Weight
Tension = Net Force + Weight
= mass × Centripetal acceleration + Weight
= 2520 kg × 9.8 m/s² + 24756 N
= 49056 N
Therefore, the tension in the cable is 49056 N.