A 2490 kg demolition ball swings at the end of a 35.9 m cable on the arc of a vertical circle. At the lowest point of the swing, the ball is moving at a speed of 7.26 m/s. Determine the tension in the cable.

Cable tension at bottom of swing =

Weight + M*V^2/R

R = 35.9 m
M = 2490 kg
V = 7.26 m/s
Weight = M*g

Well, if I were the demolition ball swinging on that cable, I would certainly be feeling a little...tense! But let's talk about the tension in the cable instead.

To find the tension in the cable, we need to consider two forces acting on the ball at the lowest point: the tension force (T) in the cable and the gravitational force (mg). At the lowest point, these two forces add up to provide the centripetal force required to keep the ball moving in a circle.

The centripetal force (F) is given by the equation:

F = mv²/r

Where:
m = mass of the ball = 2490 kg
v = velocity of the ball = 7.26 m/s
r = radius of the circle (length of the cable) = 35.9 m

So, plugging in the values:

F = (2490 kg)(7.26 m/s)² / 35.9 m

Now, we can equate this to the sum of the tension force and the gravitational force:

F = T + mg

Substituting the values:

(2490 kg)(7.26 m/s)² / 35.9 m = T + (2490 kg)(9.8 m/s²)

Now, let me unclown myself for a moment and do the math for you:

T ≈ 25731 N

So, the tension in the cable at the lowest point of the swing is approximately 25731 Newtons. But remember, tension can be a real pain sometimes, so be careful!

To determine the tension in the cable, we need to consider the forces acting on the demolition ball at the lowest point of the swing. At this point, the ball is moving horizontally, and the tension in the cable provides the centripetal force for the circular motion.

We can use the centripetal force formula to calculate the tension:

F = m * v^2 / r

where,
F = centripetal force (tension in the cable)
m = mass of the ball (2490 kg)
v = velocity of the ball (7.26 m/s)
r = radius of the circular path (length of the cable, 35.9 m).

Substituting the given values into the equation:

F = 2490 kg * (7.26 m/s)^2 / 35.9 m

F = 2490 kg * 52.8276 m^2/s^2 / 35.9 m

F = 377371.7644 N / 35.9 m

F ≈ 10518.39 N

Therefore, the tension in the cable is approximately 10518.39 N.

To determine the tension in the cable, we need to consider the forces acting on the demolition ball at the lowest point of the swing.

At the lowest point of the swing, the ball is moving in a vertical circle. The gravitational force acts downwards (towards the center of the Earth), while the tension force in the cable acts upwards (away from the center of the circle).

From the problem statement, we are given the mass of the demolition ball, m = 2490 kg, the radius of the circle, r = 35.9 m, and the speed of the ball at the lowest point, v = 7.26 m/s.

First, we need to find the acceleration of the ball at the lowest point using the centripetal acceleration formula:

a = v^2 / r

Substituting the given values, we get:

a = (7.26 m/s)^2 / 35.9 m
a ≈ 1.4727 m/s^2

Next, we can calculate the net force acting on the ball at the lowest point, which is the vector sum of the tension force and the gravitational force. Since they act in opposite directions, we subtract their magnitudes:

net force = Tension force - Gravitational force

The gravitational force can be calculated using the formula:

Gravitational force = mass * acceleration due to gravity

The acceleration due to gravity is approximately 9.8 m/s^2.

Gravitational force = mass * acceleration due to gravity
Gravitational force = 2490 kg * 9.8 m/s^2
Gravitational force ≈ 24342 N

Now, we can write down the equation for the net force:

net force = Tension force - Gravitational force

Since the net force is given by F = mass * acceleration, we can rewrite the equation as:

mass * acceleration = Tension force - Gravitational force

Substituting the known values:

2490 kg * 1.4727 m/s^2 = Tension force - 24342 N

Now, solve the equation for the tension force:

Tension force = 2490 kg * 1.4727 m/s^2 + 24342 N

After substituting the values and performing the calculation, we can determine the tension force in the cable.