Problem 89 - Physics

Question

The box of a well-known breakfast cereal states that one ounce of the cereal contains 110 Calories \((1\) food Calorie \(=4186 \mathrm{~J})\). If \(2.0 \%\) of this energy could be converted by a weight lifter's body into work done in lifting a barbell, what is the heaviest barbell that could be lifted a distance of \(2.1 \mathrm{~m}\) ?

Step-by-Step Solution

Answer

The heaviest barbell is approximately 447.48 kg.

Step 1: Convert Calories to Joules

Convert 110 Calories to Joules using the conversion factor: \[ 1 \text{ Calorie} = 4186 \text{ J} \]Thus, \[ 110 \text{ Calories} = 110 \times 4186 = 460460 \text{ J} \]

Step 2: Calculate Usable Energy for Work

Find the portion of energy that can be converted to work. Since only 2.0%, or 0.02, of the energy can be converted:\[ \text{Usable Energy} = 460460 \times 0.02 = 9209.2 \text{ J} \]

Step 3: Use the Work Formula

Use the work formula where the work done is equal to the force times the distance:\[ \text{Work} = F \cdot d \]

Step 4: Solve for Force (Weight of Barbell)

Rearrange the equation to solve for the force \( F \), where \( d = 2.1 \) m:\[ F = \frac{\text{Work}}{d} = \frac{9209.2}{2.1} \approx 4385.33 \text{ N} \]

Step 5: Convert Force to Mass

Since \( F = m \cdot g \) and \( g \approx 9.8 \text{ m/s}^2 \), rearrange to find the mass \( m \):\[ m = \frac{F}{g} = \frac{4385.33}{9.8} \approx 447.48 \text{ kg} \]

Key Concepts

Energy Conversion

Energy conversion is a fundamental principle in physics. It describes the process of changing one form of energy into another.
In our context, a weightlifter transforms chemical energy from food into mechanical energy used to lift weights.
Energy comes in many forms, such as kinetic, potential, thermal, and chemical.

• Chemical energy, like Calories from food, must be converted into usable mechanical energy.
• Conversion efficiency affects how much energy turns into work.

Only part of the energy consumed through food can be converted into actual work done.
Here, the exercise assumes a 2% conversion efficiency from Calories ingested to energy used for work.

Work and Energy

Work and energy are closely related concepts in physics. Work is defined as the amount of energy transferred by a force acting over a distance.
The formula to calculate work is:\[\text{Work} = F \cdot d\]where:

• \( F \) is the force applied, and
• \( d \) is the distance moved in the direction of the force.

The energy used to perform work is measured in Joules (J).
When a weightlifter lifts a barbell, they are doing work against gravitational force.
This exercise calculates how much work a person can potentially perform when lifting a barbell a certain height, given an initial amount of energy in Calories converted to Joules.

Force and Motion

Force and motion are interconnected concepts in physics. Force refers to any interaction that changes the motion of an object, often measured in Newtons (N).
According to Newton's Second Law of Motion, the force required to move an object is equal to the mass of the object multiplied by its acceleration:\[F = m \cdot a\]When lifting a barbell, the force needed correlates with the weight of the barbell and the acceleration due to gravity, \(g\), which is approximately \(9.8 \text{ m/s}^2\).
The calculation within this problem determines the maximum force (weight of the heaviest barbell) achievable based on the given usable energy for the lifting work over a specific distance.

Unit Conversion

Unit conversion is crucial in solving physics problems accurately.
This involves changing from one unit of measurement to another to match the needed equation or system.

• For energy, we convert from Calories to Joules, knowing that \(1 \text{ Calorie} = 4186 \text{ Joules}\).
• Force and mass conversions may be needed, especially when using units like Newtons (N) and kilograms (kg), common in physics.

In the exercise, converting the 110 Calories from cereal to Joules was necessary to compute usable energy, applicable in equations solving for force and mass.
This helps ensure consistency in calculations, allowing all parts of the problem to align properly for clear solutions.