A small blimp is used for advertising purposes at a football...

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)?

Key Concepts

Equilibrium in Constant Velocity Motion

An object moving at a constant velocity is in a state of dynamic equilibrium, meaning that the net force acting on it in every direction is zero. This implies that all forces, such as applied forces, resistive forces like drag, and gravitational and buoyant forces, balance each other out. Understanding this principle simplifies the analysis of mechanical systems by allowing one to equate the component forces to solve for unknowns like tension or work done.

Force Components

When a force is applied at an angle, it can be resolved into horizontal and vertical components using trigonometric functions like sine and cosine. This decomposition is crucial because many problems in mechanics involve forces acting in different directions, and analyzing the system often requires balancing these components separately, especially in systems with constraints such as fixed heights.

Work

Work is the measure of energy transferred when a force is applied over a displacement. It is calculated as the dot product of the force vector and the displacement vector, meaning only the component of the force in the direction of the displacement contributes to the work done. This concept is fundamental when determining the impact of forces such as tension in mechanical systems.

Drag Force

Drag force is a resistive force that acts opposite to the direction of an object's motion as it moves through a fluid, like air. It is often modeled by the equation F = K · v², where K is the drag coefficient and v is the velocity. This concept is essential in problems involving moving objects because it quantifies how air resistance affects the net force and energy required for motion.

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