Problem 23 - College Physics

Question

A 75.0 -kg athlete performs a single-hand handstand. If the area of the hand in contact with the floor is \(125 \mathrm{~cm}^{2}\) what pressure is exerted on the floor?

Step-by-Step Solution

Answer

The pressure exerted on the floor is 58,860 Pa.

Step 1: Understand Pressure Formula

Pressure is defined as the force applied perpendicular to the surface of an object per unit area over which that force is distributed. The formula for pressure \( P \) is given by: \( P = \frac{F}{A} \) where \( F \) is the force applied and \( A \) is the area.

Step 2: Calculate the Force Due to Weight

The force \( F \) exerted by the athlete is due to the gravitational pull on their body. This can be calculated using the formula \( F = m \, g \), where \( m \) is the mass of the athlete (75 kg) and \( g \) is the acceleration due to gravity \( (9.81 \, \text{m/s}^2) \). So, \( F = 75 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 735.75 \, \text{N} \).

Step 3: Convert Area to Appropriate Units

The area of the hand is given as \( 125 \, \text{cm}^2 \). To use it in the pressure formula, convert this to square meters: \( 125 \, \text{cm}^2 \times \frac{1 \, ext{m}^2}{10000 \, ext{cm}^2} = 0.0125 \, \text{m}^2 \).

Step 4: Substitute and Calculate Pressure

Now, substitute the values of force \( F \) and area \( A \) into the pressure formula: \( P = \frac{735.75 \, \text{N}}{0.0125 \, \text{m}^2} = 58860 \, \text{Pa} \). This is the pressure exerted on the floor by the hand.

Key Concepts

Force and Area

One of the foundational concepts in understanding pressure is knowing how force and area interact with each other. Force refers to any interaction that, when unopposed, changes the motion of an object. It can be a push, pull, or any action that causes an object to be displaced.
In the given scenario, the force is the gravitational pull on the athlete, which is exerted through their hand.
Area refers to the surface over which the force is spread. An important thing to note is that the larger the area over which a force is applied, the less pressure is exerted. Conversely, the smaller the area, the more concentrated and therefore greater the pressure.

• A bigger area means less focused pressure.
• A smaller area means more concentrated pressure.

Conversion of Units

When performing calculations in physics, it's crucial to convert all measurements to the correct units. Different systems of units can be confusing, but using consistent units is essential for accurate calculations.
In our scenario, the area of the hand contacts the floor is initially given in square centimeters. However, to use the pressure formula, it must be converted to square meters as the standard unit of area in physics.

• 1 square meter is equal to 10,000 square centimeters. So:
• 125 cm² × \( \frac{1 \, ext{m}^2}{10,000 \, ext{cm}^2} \) = 0.0125 m².

It is straightforward, but getting the units right ensures the answer is correct.

Gravitational Force

Gravitational force is a natural phenomenon by which all objects with mass attract one another.
On Earth, this force gives weight to physical objects and causes them to fall toward the ground when dropped.
In this exercise, the athlete's mass is key to determining the gravitational force exerted through their hand during the handstand.
The force due to gravity is calculated by multiplying the mass of the object by the gravitational acceleration constant, which is approximately 9.81 m/s² on Earth.

• For the athlete with a mass of 75 kg, the gravitational force is:
• \( F = m \times g = 75 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 735.75 \, \text{N} \).

This force acts perpendicularly to the area of contact.

Pressure Formula in Physics

The pressure formula in physics helps us determine how much force is applied over a given area. Pressure, denoted as \( P \), is defined as the force \( F \) per unit area \( A \). The formula: \[ P = \frac{F}{A} \]is simple yet powerful. It conveys that pressure increases as force increases or as area decreases.
In the athlete's handstand scenario:

• The calculated force due to gravity is 735.75 N.
• The area over which this force acts is 0.0125 m².
• Inserting these into the formula: \( P = \frac{735.75 \, \text{N}}{0.0125 \, \text{m}^2} = 58860 \, \text{Pa} \).

Thus, 58,860 pascals (Pa) of pressure is exerted on the floor by the athlete's hand.