91Ó°ÊÓ

Kinetic energy is the energy that an object has due to its motion. The formula for kinetic energy is given by \( K = \frac{1}{2} mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the object. In this exercise, the box initially has no kinetic energy because its starting velocity is zero.
As the box is pushed up the ramp and gains velocity, its kinetic energy increases. By the time the box reaches the top of the ramp, its velocity is \( 0.80 \, \mathrm{m/s} \), resulting in a kinetic energy of \( 0.736 \, \mathrm{J} \).
To find the change in kinetic energy (\( \Delta K \)), we subtract the initial kinetic energy from the final kinetic energy. Since the initial kinetic energy is zero, \( \Delta K \) equals \( 0.736 \, \mathrm{J} \).
Understanding how velocity and mass affect kinetic energy is important for analyzing motion and energy conservation in physical systems.

Potential energy is the energy stored in an object due to its position, height, or arrangement. Often, in physics, we focus on gravitational potential energy, which arises due to an object's position relative to Earth. The formula to calculate gravitational potential energy is \( U = mgh \), where \( m \) is the mass, \( g \) is the gravitational acceleration, and \( h \) is the height.
In the scenario given, the box gains gravitational potential energy as it is pushed up the ramp. The height \( h \) the box is raised can be determined by multiplying the distance moved along the ramp with the sine of the ramp's angle \( ( \sin(17^\circ) ) \). This results in a height of \( 0.574 \, \mathrm{m} \), giving us a final gravitational potential energy of \( 12.89 \, \mathrm{J} \).
Initially at the base of the ramp, the box has zero potential energy because its height is zero. Therefore, the change in potential energy \( \Delta U_g \) is equal to the final potential energy, \( 12.89 \, \mathrm{J} \). Remembering that potential energy changes with height helps in understanding energy transformations in elevated systems.

Thermal energy refers to the internal energy present in a system due to the random motion of its particles. In physics problems, thermal energy often becomes significant when energy is transformed from mechanical forms to heat via friction or resistance.
In this particular exercise, as the box is pushed up the ramp, some of the work done is transformed into thermal energy due to friction along the surface. To find the change in thermal energy (\( \Delta E_{\mathrm{th}} \)), we use the work-energy theorem, factoring in all changes in kinetic and potential energy, and the work done by the force.
The work done on the system is \( 20 \, \mathrm{J} \). By subtracting the change in kinetic energy \( (0.736 \, \mathrm{J}) \) and potential energy \( (12.89 \, \mathrm{J}) \) from the total work done, we find that \( \Delta E_{\mathrm{th}} \) is \( 6.37 \, \mathrm{J} \). Keep in mind that thermal energy changes often reduce the mechanical energy available in systems involving friction.

The work-energy theorem is a crucial concept in mechanics, connecting the work done on an object to its kinetic energy change. Specifically, the theorem states that the work done by all forces acting on a system equals the change in kinetic energy of that system (\( W = \Delta K \)).
In the given problem, the theorem helps us understand how the force applied to push the box does work on the system and causes changes in kinetic and potential energy. The work-energy theorem can also be adapted to account for other energy forms, such as thermal energy, especially in cases where non-conservative forces like friction are present.
By examining the total work (\( 20 \, \mathrm{J} \)) alongside changes in kinetic energy \( (0.736 \, \mathrm{J}) \) and potential energy \( (12.89 \, \mathrm{J}) \), we see how the remainder becomes thermal energy (\( 6.37 \, \mathrm{J} \)) due to friction. This theorem is a powerful tool for analyzing energy transformations and ensuring energy conservation across physical processes.