Question

A small blimp is used for advertising purposes at a football game. It has a mass of \(93.5 \mathrm{~kg}\) and is attached by a towrope to a truck on the ground. The towrope makes an angle of \(53.3^{\circ}\) downward from the horizontal, and the blimp hovers at a constant height of \(19.5 \mathrm{~m}\) above the ground. The truck moves on a straight line for \(840.5 \mathrm{~m}\) on the level surface of the stadium parking lot at a constant velocity of \(8.90 \mathrm{~m} / \mathrm{s}\). If the drag coefficient \(\left(K\right.\) in \(\left.F=K v^{2}\right)\) is \(0.500 \mathrm{~kg} / \mathrm{m}\), how much work is done by the truck in pulling the blimp (assuming there is no wind)?

Short answer

Question: Calculate the work done by the truck in pulling the blimp through a specific distance, given the following information: - Mass of the blimp = 93.5 kg - Acceleration due to gravity = 9.81 m/s² - Angle of the towrope = 53.3 degrees - Drag coefficient = 0.500 kg/m - Velocity of the blimp = 8.90 m/s - Distance the truck moved = 840.5 m Answer: The work done by the truck in pulling the blimp is approximately 625,832.61 Joules.

Step-by-step solution

Find the tension in the towrope.

To find the tension in the towrope, we first need to find the gravitational force acting on the blimp. The gravitational force can be calculated using the formula: \(F_g = m \times g\) Where \(F_g\) is the gravitational force, \(m\) is the mass of the blimp (93.5 kg), and \(g\) is the acceleration due to gravity (approximately \(9.81 \mathrm{~m} / \mathrm{s^2}\)). Now calculate the gravitational force: \(F_g = 93.5 \times 9.81 = 916.35 \mathrm{~N}\) Next, since the blimp is hovering at a constant height, the vertical component of the tension in the towrope must be equal to the gravitational force. So we can use trigonometry to find the tension in the towrope: \(T \sin(53.3) = F_g\) Now solve for T: \(T = \frac{F_g}{\sin(53.3)} = \frac{916.35}{\sin(53.3)} = 1186.62 \mathrm{~N}\)

Calculate the force exerted by the truck on the blimp.

The force exerted by the truck on the blimp is equal to the horizontal component of the tension in the towrope. Therefore: \(F_{truck} = T \cos(53.3) = 1186.62 \times \cos(53.3) = 780.25 \mathrm{~N}\)

Calculate the drag force.

The drag force can be calculated using the drag force equation: \(F_{drag} = K \times v^2\) Where \(F_{drag}\) is the drag force, \(K\) is the drag coefficient (0.500 kg/m), and \(v\) is the velocity of the blimp (8.90 m/s). Now calculate the drag force: \(F_{drag} = 0.5 \times (8.90)^2 = 35.43 \mathrm{~N}\)

Calculate the net force on the blimp.

The net force on the blimp is the difference between the force exerted by the truck and the drag force: \(F_{net} = F_{truck} - F_{drag} = 780.25 - 35.43 = 744.82 \mathrm{~N}\)

Calculate the work done by the truck on the blimp.

Now we can use the work-energy theorem to calculate the work done by the truck on the blimp: \(W = F_{net} \times d\) Where \(W\) is the work done, \(F_{net}\) is the net force on the blimp (744.82 N), and \(d\) is the distance the truck moved (840.5 m). Calculate the work done: \(W = 744.82 \times 840.5 = 625832.61 \mathrm{~J}\) The work done by the truck in pulling the blimp is approximately \(625,832.61 \mathrm{~J}\) (Joules).

Key concepts

Work-Energy Theorem

The Work-Energy Theorem is a vital concept in physics that links work and energy. It states that the work done on an object is equal to the change in its kinetic energy. This is a powerful tool, allowing us to understand how forces affect an object's speed and motion. Specifically, in the given exercise, the theorem helps us calculate the work done by the truck in pulling the blimp over a specific distance. The formula used is:
\[ W = F_{net} \times d \]

• \( W \) is the work done on the object.
• \( F_{net} \) is the net force acting on the object.
• \( d \) is the distance over which the force is applied.

The net force is calculated by subtracting the drag force from the force exerted by the truck. The work done on the blimp tells us how much energy is transferred due to the forces acting along the distance traveled.

Drag Force

Drag Force is a resistance force caused by the motion of an object through a fluid, like air or water. It opposes the direction of the object's movement and depends on factors such as velocity and the drag coefficient.
The drag force equation for our exercise is:
\[ F_{drag} = K \times v^2 \]

• \( F_{drag} \) is the drag force resisting the motion.
• \( K \) is the drag coefficient (for the blimp, it is 0.500 kg/m).
• \( v \) is the velocity of the moving object (8.90 m/s for the blimp).

In the exercise, calculating the drag helps determine how much resistance the blimp faces while being pulled. This is crucial to find the net force acting on the blimp for further calculations with the Work-Energy Theorem.

Tension in Towrope

Tension in the towrope is a key player in pulling the blimp. It is the force transmitted through the rope connecting the blimp to the truck. Here, the tension must counteract the gravitational pull on the blimp and include a horizontal component that aids in towing the blimp.
To find the tension in the rope, we use trigonometry:
\[ T \sin(53.3) = F_g \]

• \( T \) is the tension in the towrope.
• \( F_g \) is the gravitational force (approximately 916.35 N).

After finding the total tension (which supports the blimp's weight), its horizontal component is calculated using:
\[ F_{truck} = T \cos(53.3) \]
This horizontal force is critical in maintaining the blimp's motion across the parking lot and helps overcome the drag force acting against it. Understanding tension is essential to ensure the blimp stays at a stable height and is smoothly towed behind the truck.