Problem 2 - University Physics with Modern Physics

Question

A tow truck pulls a car 5.00 \(\mathrm{km}\) along a horizontal roadway using a cable having a tension of 850 \(\mathrm{N}\) . (a) How much work does the cable do on the car if it pulls horizontally? If it pulls at \(35.0^{\circ}\) above the horizontal? (b) How much work does the cable do on the tow truck in both cases of part (a)? (c) How much work does gravity do on the car in part (a)?

Step-by-Step Solution

Answer

(a) 4250 kJ, 3480.75 kJ; (b) -4250 kJ, -3480.75 kJ; (c) 0 J.

Step 1: Understand Work Calculation

Work done is computed with the formula: \[ W = F imes d imes \cos(\theta) \]where:- \(W\) is the work done,- \(F\) is the force applied (850 N in this case),- \(d\) is the distance over which the force is applied (5.00 km = 5000 m),- \(\theta\) is the angle between the force direction and the direction of movement.

Step 2: Calculate Work for Horizontal Pull

When the cable pulls horizontally, \( \theta = 0^{\circ} \). Thus, the work done by the cable on the car is:\[ W = 850 \times 5000 \times \cos(0^{\circ}) \]\[ W = 850 \times 5000 \times 1 \]\[ W = 4250000 \, \mathrm{J} \]

Step 3: Calculate Work for Pull at 35 Degrees

When the cable pulls at \(35^{\circ}\) above the horizontal, the work done is:\[ W = 850 \times 5000 \times \cos(35^{\circ}) \]Using \( \cos(35^{\circ}) \approx 0.819 \):\[ W = 850 \times 5000 \times 0.819 \]\[ W \approx 3480750 \, \mathrm{J} \]

Step 4: Understand Work on Tow Truck

The work done on the tow truck by the cable will be equal in magnitude but opposite in direction (negative) since the movement is the same.

Step 5: Calculate Work on Tow Truck for Horizontal and 35 Degrees

- For horizontal pull, the work done on the tow truck is: \[ W = - 4250000 \, \mathrm{J} \]- For pull at \(35^{\circ}\): \[ W = - 3480750 \, \mathrm{J} \]

Step 6: Evaluate Work Done by Gravity

Gravity does no work since the motion is horizontal and perpendicular to the force of gravity. Thus, work done by gravity is 0 J.

Key Concepts

Physics Problems

In physics, when dealing with problems like the one involving a tow truck and a car, it's essential to dissect the situation into manageable components. This approach helps in understanding the principles at play, such as forces and motion.
Physics problems often involve concepts like work and energy, which are central to understanding how force moves an object over a distance. The principle guiding these problems is the work-energy principle.

• Work is the energy transferred by a force over a distance in the direction of the force.
• Energy concepts help describe whether a task was physically taxing or required significant effort.

In our specific problem, we're analyzing the work done as a car is pulled horizontally by a tow truck. Understanding the way forces interact in such scenarios allows for the resolution of various physics challenges.

Force and Tension

Force is a vector quantity, which means it has both magnitude and direction. When a cable pulls a car, the tension in the cable is the force exerted. Tension is the pulling force transmitted axially by the string or cable.

• Tension can be visualized as the cables being taut under a pulling force.
• It's measured in newtons (N) and affects how work is calculated in our exercise.

In this scenario, the tension is 850 N, directly influencing the work done on the car. Understanding force and tension helps in predicting how objects will accelerate or resist the force applied, essential in solving physics problems.

Angle of Application

The angle of application refers to the angle at which a force is applied concerning a reference direction, often the horizontal surface in this context. The importance of the angle is shown in the equation for work: \[ W = F \times d \times \cos(\theta) \] The angle, \(\theta\), determines how much of the force effectively contributes to moving the object in the desired direction.

• At \(\theta = 0^{\circ}\), all the force contributes to the movement.
• At \(35^{\circ}\), only a part of the force does, as represented by the cosine of the angle.

This specificity affects the total work done and ultimately demonstrates how work depends not just on force and distance, but also on direction.

Horizontal Motion

Horizontal motion is straightforward when considering displacement along a flat plane. This motion is critical in the exercise, where a car is pulled by a truck along a horizontal road.

• All forces acting vertically, like gravity, do not perform work in purely horizontal motion.
• This scenario allows us to focus on the horizontal component of the force exerted.

By understanding horizontal motion, one can see that any vertical forces, like gravity in this situation, are perpendicular to the movement. As such, they do not contribute to the work done, simplifying our calculations drastically.