Question

The following event actually occurred on the Sunshine Skyway Bridge near St. Petersburg, Florida, in \(1997 .\) Five daredevils tied a \(55-\mathrm{m}\) -long cable to the center of the bridge. They hoped to swing back and forth under the bridge at the end of this cable. The five people (total weight \(=W\) ) attached themselves to the end of the cable, at the same level and \(55 \mathrm{~m}\) away from where it was attached to the bridge and dropped straight down from the bridge, following the dashed circular path indicated in the figure. Unfortunately, the daredevils were not well versed in the laws of physics, and the cable broke (at the point it was linked to their seats) at the bottom of their swing. Determine how strong the cable (and all the links where the seats and the bridge are attached to it) would have had to be in order to support the five people at the bottom of the swing. Express your result in terms of their total weight, \(W\).

Short answer

Answer: The maximum force the cable must endure to support the five daredevils at the bottom of their swing can be expressed as: \(F_c = \frac{2Wh}{r}\), where \(F_c\) is the centripetal force, \(W\) is the total weight of the daredevils, \(h\) is the height of the swing, which is equal to the length of the cable, and \(r\) is the radius of the circular path, which is also equal to the length of the cable.

Step-by-step solution

Identify both potential and kinetic energies at the initial and final points.

Initially, the daredevils are at their highest point at the edge of the bridge, with all their potential energy, and they will swing down to the bottom, where all their energy will be converted to kinetic energy. Let's call the initial point of the swing A and the bottom point of the swing B.

Set up the conservation of mechanical energy equation.

According to the conservation of mechanical energy, the total mechanical energy at points A and B remains the same. Therefore, we have: \(E_A = E_B\) where \(E_A\) is the mechanical energy at point A and \(E_B\) is the mechanical energy at point B.

Write down the expressions for the potential and kinetic energies.

The potential energy at point A is: \(PE_A = mgh\). The potential energy at point B is zero. The kinetic energy at point A is zero. The kinetic energy at point B is: \(KE_B = \frac{1}{2}mv^2\), where \(m\) is the total mass of the five daredevils, \(g\) is the gravitational acceleration constant, \(h\) is the height of the swing, which is equal to the length of the cable (\(55m\)), and \(v\) is the velocity at point B.

Substitute the values for the potential and kinetic energies in the conservation of mechanical energy equation.

Now, we can substitute the values into the conservation of mechanical energy equation, which will give us: \(mgh = \frac{1}{2}mv^2\)

Find the velocity at point B.

We can solve for the velocity at the bottom of the swing: \(v^2 = 2gh\) \(v = \sqrt{2gh}\)

Determine the centripetal force at point B.

At the bottom of the swing, the daredevils experience a centripetal force due to their circular motion. This force can be calculated as: \(F_c = m\frac{v^2}{r}\) where \(F_c\) is the centripetal force, and \(r\) is the radius of the circular path (which is equal to the length of the cable \(55m\)).

Substitute the velocity expression into the centripetal force equation.

Now we can substitute the velocity expression from step 5 into the centripetal force equation: \(F_c = m\frac{(2gh)}{r}\)

Express centripetal force in terms of the total weight.

Notice that \(mg=W\), which is the total weight of the daredevils. So, we can now express the centripetal force in terms of the total weight: \(F_c = \frac{2Wh}{r}\) This is how much force the cable must endure as the daredevils reach the bottom of their swing. The cable (and all the links where the seats and the bridge are attached to it) would have had to be strong enough to withstand this force for all daredevils not to fall.

Key concepts

Conservation of Mechanical Energy

The principle of conservation of mechanical energy is a fundamental concept in physics that states that the total mechanical energy in a closed system remains constant, as long as only conservative forces are acting.

When we consider the daredevils' thrilling adventure on the Sunshine Skyway Bridge, the system is composed of the people tied to the cable and Earth's gravitational field. At the beginning of their descent, at the top of the bridge, they have maximum potential energy and zero kinetic energy because they are not moving. As they fall, this potential energy is transformed into kinetic energy, causing them to accelerate, until right before the cable snaps where their speed is at its maximum and potential energy is at its lowest, essentially zero at the lowest point of the swing.

Mathematically, this energy transformation without any energy loss is expressed as:
\(E_A = PE_A\), which is equal to the kinetic energy at the bottom, \(E_B = KE_B\). The formula \(PE_A = mgh\) represents the potential energy at the start, and \(KE_B = \frac{1}{2}mv^2\) the kinetic energy at the bottom, where \(h\) is the height of the swing.

Kinetic Energy and Potential Energy Equations

Kinetic energy and potential energy are the two primary forms of mechanical energy. The kinetic energy of an object is due to its motion and is given by the equation \(KE = \frac{1}{2}mv^2\), where \(m\) is the mass and \(v\) is the velocity of the object. On the other hand, potential energy is energy stored in an object due to its position or configuration.

For the daredevils swinging under the bridge, at the highest point (Point A), their potential energy is maximized and expressed by \(PE = mgh\), with \(g\) standing for the acceleration due to gravity and \(h\) representing the height above the reference point.

As they swung down, they lost height but gained speed. At the bottom of the swing, their height is zero, leading to zero potential energy and maximum kinetic energy, which was calculated to find their velocity at the bottom of the swing before the cable broke.

The exercise required understanding the interchange between potential energy and kinetic energy during the motion, which was crucial in determining the cable's necessary strength.

Circular Motion in Physics

Circular motion is a type of motion where an object moves along a curved path, such as a circle. The direction of the velocity vector constantly changes, meaning the object is in a state of acceleration, even if it moves at a constant speed. This acceleration is directed towards the center of the circular path and is called centripetal acceleration.

The force that is responsible for this centripetal acceleration is called the centripetal force. In the case of the daredevils, they provided the centripetal force through the tension in the cable while swinging in a circular path beneath the bridge. The centripetal force on an object moving in a circle of radius \(r\) with velocity \(v\) is given by \(F_c = m\frac{v^2}{r}\). This is the force that keeps them moving in a circular path rather than flying off in a straight line due to inertia.

In the exercise, the centripetal force calculation revealed how much strength the cable needed to have to support the weight of the five daredevils at the critical point at the bottom of the swing, where the centripetal force was highest.