Problem 18 - College Physics

Question

In a college homecoming competition, eighteen students lift a sports car. While holding the car off the ground, each student exerts an upward force of \(400 \mathrm{~N}\). (a) What is the mass of the car in kilograms? (b) What is its weight in pounds?

Step-by-Step Solution

Answer

(a) The mass of the car is approximately 734.69 kg. (b) Its weight is approximately 1620 lbs.

Step 1: Calculate the Total Lifting Force

To find the total lifting force exerted by all the students, multiply the force each student exerts by the number of students: \[F_{\text{total}} = 18 \times 400 \text{ N} = 7200 \text{ N}\]

Step 2: Convert Force to Mass

Use the formula relating force, mass, and acceleration due to gravity: \[F = mg\]Solve for mass \(m\) by dividing the total force by the acceleration due to gravity \(g = 9.8 \text{ m/s}^2\):\[m = \frac{F}{g} = \frac{7200 \text{ N}}{9.8 \text{ m/s}^2} \approx 734.69 \text{ kg}\]

Step 3: Convert Mass to Weight in Pounds

The weight in pounds can be found by first converting the mass from kilograms to pounds. Use the conversion factor 1 kg = 2.20462 lbs:\[\text{Weight} = 734.69 \text{ kg} \times 2.20462 \text{ lbs/kg} \approx 1620 \text{ lbs}\]

Key Concepts

Force and Mass

Force and mass are fundamental concepts in physics closely related to how objects interact with each other. Force is generally defined as a push or pull on an object resulting from the object's interaction with another object. It is measured in newtons ( N ), with one newton being the force required to accelerate a one-kilogram mass by one meter per second squared.  

• Force ( F ): A vector quantity, meaning it has both magnitude and direction.
• Mass ( m ): A scalar quantity representing the amount of matter in an object, measured in kilograms ( kg ).

In the exercise, eighteen students exert a force on a sports car. By knowing the total force they've exerted, we can determine the mass. By calculating how much each student contributes, you multiply the individual force by the number of students ( 18 imes 400 N = 7200 N ), providing the total upward force on the car. This total force allows us to solve for the car's mass using the relationship between force, mass, and gravity.

Newton's Laws

Newton's Laws of Motion are a set of three fundamental principles essential for understanding the relationship between a body and forces acting upon it. These laws offer insights into how forces result in motion or change in an object's state. 

• First Law: An object remains at rest or in uniform motion unless acted upon by an external force.
• Second Law: The force acting on an object is equal to the mass of that object times its acceleration ( F=ma ).
• Third Law: For every action, there is an equal and opposite reaction.

In this problem, we primarily use Newton's Second Law, as it relates force, mass, and acceleration. When 7200 N is applied to the car by the students (force), and knowing the constant acceleration due to gravity ( g = 9.8 ext{ m/s}^2 ), we can calculate the car's mass by rearranging the formula to m = F/g . This showcases how an external force impacts a car's state of rest by enabling us to lift it.

Unit Conversion

Unit conversion is the process of changing a measure of physical quantity from one unit to another without affecting its value. It is essential for interpreting and communicating scientific measurements universally. In physics, the known and accepted units might differ based on geographical locations, like the metric system compared to the imperial system. In our exercise:

• Converting Newtons to Kilograms: Using the relation F = mg , we divide the force by g=9.8 ext{ m/s}^2 to find mass in kilograms.
• Converting Kilograms to Pounds: Since 1 kilogram equals approximately 2.20462 pounds, multiply the mass in kg by this conversion factor ( 734.69 ext{ kg} imes 2.20462 ext{ lbs/kg} ≈ 1620 lbs dealing with the car's weight).

Using these conversions ensures that the results are useful across different systems of measurement, helping make physics problems universally understandable and solutions versatile.

Gravity

Gravity is one of the four fundamental forces of nature, though it is the weakest at the particle level. It is a crucial factor on a macroscopic scale, keeping planets in orbit and causing objects to fall toward the Earth. On Earth, gravity's effect is a constant acceleration denoted by g = 9.8 ext{ m/s}^2 . It represents the rate at which an object accelerates towards the Earth when dropped, assuming no other forces act on it. In physics equations, gravity is often represented as g when calculating weight or the force of gravity on an object.

• Weight and Gravity: Weight is the force of gravity acting on an object's mass, calculated using the formula F = mg , where F represents weight in newtons.
• Impact on Calculations: By knowing gravity's consistent acceleration, we can easily convert force to mass and vice-versa, using the gravitational constant in calculations.

In the given problem, gravity allows us to transition the measured upward force into a calculable mass of the car, showing how gravity underpins how we perceive weight and mass relationships in everyday phenomena and exercises.