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Chapter 1: Problem 34

A standard interior staircase has steps each with a rise (height) of \(19 \mathrm{~cm}\) and a run (horizontal depth) of \(23 \mathrm{~cm}\). Research suggests that the stairs would be safer for descent if the run were, instead, \(28 \mathrm{~cm}\). For a particular staircase of total height \(4.57 \mathrm{~m}\), how much farther would the staircase extend into the room at the foot of the stairs if this change in run were made?

Short Answer

Expert verified
The staircase would extend an additional 120 cm (1.2 meters) into the room.

Step by step solution

01

- Convert the total height of the staircase

Convert the total height of the staircase from meters to centimeters: Multiply the height in meters by 100 to get the height in centimeters. Given: 4.57 meters4.57 meters * 100 = 457 centimeters
02

- Calculate the number of steps

Divide the total height of the staircase by the rise (height) of each step to determine the number of steps.Given: rise = 19 cmNumber of steps = Total height / RiseNumber of steps = 457 cm / 19 cm per step 鈮� 24.05 stepsSince the number of steps must be a whole number, round to the nearest whole number, 24.
03

- Calculate the original total run

Multiply the number of steps by the original run (horizontal depth) per step to find the original floor space taken by the staircase.Original run per step = 23 cmOriginal total run = Number of steps * Original runOriginal total run = 24 * 23 cm = 552 cm
04

- Calculate the new total run with the safer run

Multiply the number of steps by the new run (horizontal depth) per step to find the new floor space taken by the staircase.New run per step = 28 cmNew total run = Number of steps * New runNew total run = 24 * 28 cm = 672 cm
05

- Determine the additional distance extended into the room

Subtract the original total run from the new total run to find how much farther the staircase would extend into the room.Additional distance = New total run - Original total runAdditional distance = 672 cm - 552 cm = 120 cm

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Key Concepts

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

Staircase Design
When it comes to architectural design, staircases play a crucial role in ensuring safe and efficient movement between different levels of a building. The primary components of a staircase include the rise (vertical height of each step) and the run (horizontal depth of each step). These dimensions significantly affect the usability and safety of the staircase.
In the given problem, the original staircase has a rise of 19 cm and a run of 23 cm. This combination determines the angle and the comfort while climbing or descending the stairs. A well-designed staircase should balance both ease of use and safety considerations.
Dimension Conversion
To solve real-life problems in physics, converting measurements into consistent units is essential. In this exercise, the total height of the staircase is given in meters, but calculations are easier to manage in centimeters to match the steps' dimensions.
Conversion is straightforward: multiply the height in meters by 100. Thus, 4.57 meters becomes 457 centimeters.
Consistent units simplify further calculations, ensuring accuracy and reducing the chances of errors.
Safe Stair Design
Safety is paramount in staircase design. Research shows that increasing the run (horizontal depth) can make stairs safer by providing a larger horizontal surface for each step. It helps prevent tripping and makes the staircase more forgiving during descent.
In this exercise, the safer design increases the run from 23 cm to 28 cm. Calculating the new extended distance into the room involves determining the total run for both original and new designs and then finding the difference. This improvement not only enhances safety but also alters the overall space utilization within the building.
Physics in Everyday Life
Physics principles are ingrained in daily activities and architectural designs. Understanding simple physics concepts like force, balance, and measurement conversion is crucial in creating safe and functional structures such as staircases. These principles help determine the ideal dimensions for usability and safety.
In the staircase problem, converting dimensions, calculating total heights, and determining the spatial impact of design changes embody practical applications of physics in everyday life. Such exercises underscore the importance of physics in making informed design choices that enhance safety and comfort.

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Most popular questions from this chapter

Speed of Light Express the speed of light, \(3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}\), in (a) feet per nanosecond and (b) millimeters per picosecond.

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In purchasing food for a political rally, you erroneously order shucked medium-size Pacific oysters (which come 8 to 12 per U.S. pint) instead of shucked medium-size Atlantic oysters (which come 26 to 38 per U.S. pint). The filled oyster container delivered to you has the interior measure of \(1.0 \mathrm{~m} \times 12 \mathrm{~cm} \times 20 \mathrm{~cm}\), and a U.S. pint is equivalent to \(0.4732\) liter. By how many oysters is the order short of your anticipated count?

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