Which equation correctly represents the mechanical energy of a system?
A. [tex]$ME =\frac{1}{2} m v^2-m g h$[/tex]
B. [tex]$ME =\frac{1}{2} m v^2+m g h$[/tex]
C. [tex]$ME =\frac{\frac{1}{2} m v^2}{m g h}$[/tex]
D. [tex]$ME =m g h-\frac{1}{2} m v^2$[/tex]
Answer (2)
Mechanical energy is the sum of kinetic and potential energy.
Kinetic energy is 2 1 m v 2 and potential energy is m g h .
Therefore, mechanical energy is ME = 2 1 m v 2 + m g h .
The correct equation is ME = 2 1 m v 2 + m g h .
Explanation
Understanding Mechanical Energy Mechanical energy (ME) is the sum of kinetic energy (KE) and potential energy (PE). Kinetic energy is given by 2 1 m v 2 , where m is mass and v is velocity. Potential energy due to gravity is given by m g h , where m is mass, g is the acceleration due to gravity, and h is height.
Expressing Mechanical Energy as a Sum The mechanical energy (ME) of a system is the sum of its kinetic energy (KE) and potential energy (PE). We can write this as: ME = K E + PE
Substituting KE and PE Formulas Substitute the formulas for kinetic and potential energy into the equation: ME = 2 1 m v 2 + m g h
Identifying the Correct Option Comparing this equation with the given options, we see that option B matches our derived equation. Therefore, the correct equation representing the mechanical energy of a system is: ME = 2 1 m v 2 + m g h
Examples
Understanding mechanical energy is crucial in designing roller coasters. The total mechanical energy (potential and kinetic) at the start of the ride determines the maximum height and speed the coaster can achieve, neglecting friction. Engineers use this principle to ensure the coaster has enough energy to complete loops and hills safely, providing an exhilarating yet secure experience for riders. By carefully balancing potential and kinetic energy, they create thrilling rides that adhere to the laws of physics.
The correct equation representing the mechanical energy of a system is option B: ME = 2 1 m v 2 + m g h . This equation is the sum of kinetic energy and potential energy. It highlights the relationship between an object's motion, mass, and its position in a gravitational field.
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