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Chapter 1: Problem 79

Perform the following calculations and express your answer using the correct number of significant digits. (a) A woman has two bags weighing 13.5 lb and one bag with a weight of 10.2 lb. What is the total weight of the bags? (b) The force \(F\) on an object is equal to its mass \(m\) multiplied by its acceleration \(a\). If a wagon with mass 55 kg accelerates at a rate of \(0.0255 \mathrm{m} / \mathrm{s}^{2}\), what is the force on the wagon? (The unit of force is called the newton and it is expressed with the symbol N.)

Short Answer

Expert verified
(a) The total weight of the three bags is \(37.2~\textrm{lb}\). (b) The force on the wagon is \(1.4~\textrm{N}\).

Step by step solution

01

(a) Adding the weights of the bags

To find the total weight of the bags, we need to add the weights of all three bags and make sure to use the correct number of significant digits. We have: - Bag 1: \(13.5~\textrm{lb}\) - Bag 2: \(13.5~\textrm{lb}\) - Bag 3: \(10.2~\textrm{lb}\) Now, let's add the weights and consider the least number of decimal places in the given numbers.
02

(a) Calculating the total weight

Adding the weights together gives the total weight: \(13.5~\textrm{lb} + 13.5~\textrm{lb} + 10.2~\textrm{lb} = 37.2~\textrm{lb}\) The total weight of the three bags is \(37.2~\textrm{lb}\), which has two significant digits.
03

(b) Using the formula for force

The force F on an object is equal to its mass m multiplied by its acceleration a. The formula for force is: \(F = m \cdot a\) We are given the mass \(m = 55~\textrm{kg}\) and the acceleration \(a = 0.0255~\textrm{m/s}^2\). We need to multiply the mass and acceleration while considering the correct number of significant digits.
04

(b) Calculating the force on the wagon

Using the formula for force, we can calculate the force on the wagon: \(F = 55~\textrm{kg} \cdot 0.0255~\textrm{m/s}^2 = 1.4025~\textrm{N}\) Now, we need to express our answer using the correct number of significant digits. We are given the mass with only two significant digits (55 kg), so our final answer should have two significant digits as well. Thus, we round 1.4025 N to 1.4 N The force on the wagon is \(1.4~\textrm{N}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Force Calculation
Force calculation is a common exercise in physics and engineering. It involves using a fundamental formula:
  • Force (\( F \)) is the product of mass (\( m \)) and acceleration (\( a \)).
  • Mathematically, the formula is: \( F = m \cdot a \).
In the exercise provided, a wagon with a mass of 55 kg accelerates at a rate of 0.0255 m/s虏. To find the force, multiply the mass by the acceleration.
  • Calculate: \( 55~\text{kg} \times 0.0255~\text{m/s}^2 = 1.4025~\text{N} \).
Round the calculated force to the appropriate number of significant figures. In this case, you have to match the least amount of significant figures given in the problem, which is two (from \( 55~\text{kg} \)). Therefore, the resulting force is rounded to \( 1.4~\text{N} \). Understanding this concept is vital for solving physics problems involving forces.
Unit Conversion
Unit conversion is a crucial skill in physics as different parameters might use varying units. Proper conversions ensure calculations are consistent and correct. For forces, understanding that:
  • The unit of force is the newton (N).
  • 1 N equals 1 kg路m/s虏.
These conversions are often necessary when working with different measurement systems, such as converting weight from pounds (lb) to kilograms (kg) and distance from feet to meters. If the given problem presents data in mixed units, always convert them to the same unit system before solving the problem. This ensures that your calculations, especially with mixed data like mass and acceleration, provide consistent physics solutions.
Physics Problem Solving
Physics problem solving requires a strategic approach.
  • First, understand the problem by identifying known and unknown quantities.
  • Use relevant formulas, like \( F = m \cdot a \) for force, understanding each symbol represents a specific concept.
  • Convert all units to ensure uniformity across the problem scope.
  • Apply the solution accurately using significant figures for final answers.
This systematic approach not only provides the correct result but also hones critical problem-solving skills crucial in physics. Practice with various scenarios to increase familiarity with concepts like force calculations and unit conversions. Remember, consistency in solving steps and correct unit usage are the keys to success in physics calculations.

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Most popular questions from this chapter

The purpose of this problem is to show the entire concept of dimensional consistency can be summarized by the old saying "You can't add apples and oranges." If you have studied power series expansions in a calculus course, you know the standard mathematical functions such as trigonometric functions, logarithms, and exponential functions can be expressed as infinite sums of the form \(\sum_{n=0}^{\infty} a_{n} x^{n}=a_{0}+a_{1} x+a_{2} x^{2}+a_{3} x^{3}+\cdots\),where \(\begin{array}{llllll}\text { the } & a_{n} & \text { are } & \text { dimensionless } & \text { constants } & \text { for } & \text { all }\end{array}\) \(n=0,1,2, \cdots\) and \(x\) is the argument of the function. (If you have not studied power series in calculus yet, just trust us.) Use this fact to explain why the requirement that all terms in an equation have the same dimensions is sufficient as a definition of dimensional consistency. That is, it actually implies the arguments of standard mathematical functions must be dimensionless, so it is not really necessary to make this latter condition a separate requirement of the definition of dimensional consistency as we have done in this section.

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State how many significant figures are proper in the results of the following calculations: (a) \((106.7)(98.2) /(46.210)(1.01)\); (b) \((18.7)^{2}\) (c) \(\left(1.60 \times 10^{-19}\right)(3712)\)
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