Question

A single conservative force \(\mathrm{F}(\mathrm{x})\) acts on a \(2.5 \mathrm{~kg}\) particle that moves along the \(\mathrm{x}\) -axis. The potential energy \(\mathrm{U}(\mathrm{x})\) is given by \(\mathrm{U}(\mathrm{x})=\left[10+(\mathrm{x}-4)^{2}\right]\) where \(\mathrm{x}\) is in meter. \(\mathrm{At} \mathrm{x}=6.0 \mathrm{~m}\) the particle has kinetic energy of \(20 \mathrm{~J}\). what is the mechanical energy of the system? (A) \(34 \mathrm{~J}\) (B) \(45 \mathrm{~J}\) (C) \(48 \mathrm{~J}\) (D) \(49 \mathrm{~J}\)

Short answer

= 10 + (2)^2 = 10 + 4 = 14 J #Step 2: Add the potential energy U to the given kinetic energy K to get the total mechanical energy E# #tag_title#Calculate the Total Mechanical Energy #tag_content#Mechanical Energy, E(x) = K + U E(6) = 20 J + 14 J = 34 J The mechanical energy of the system is \(34 J\). Answer: \(\boxed{(A)}\).

Step-by-step solution

Find the Potential Energy at x = 6.0 m

Given the potential energy function U(x) = [10 + (x - 4)^2], substitute x = 6.0 m into the function to calculate the potential energy at x = 6.0 m: U(6) = [10 +(6 - 4)^2]

Key concepts

Potential Energy

Potential energy is the stored energy in an object due to its position or state. In this exercise, the potential energy of a particle is determined by its position along the x-axis.
The potential energy function is given by: \[ U(x) = [10 + (x-4)^2] \] This equation shows how the potential energy changes with the particle's location.
When the particle is at x = 6.0 m, we can plug this value into the function to find its potential energy:

• Substituting, we have: \[ U(6) = [10 + (6-4)^2] = [10 + (2)^2] \]
• Simplifying further, \[ U(6) = 10 + 4 = 14 \, \text{J} \]

Thus, at x = 6.0 m, the stored energy of the particle is 14 joules.

Kinetic Energy

Kinetic energy refers to the energy an object possesses due to its motion. It depends on two factors: the mass of the object and its velocity.
The kinetic energy (KE) can be calculated using the formula: \[ KE = \frac{1}{2}mv^2 \] In this scenario, we are told that the particle already has kinetic energy of 20 J at x = 6.0 m.

• The particle is moving along the x-axis which contributes to its kinetic energy.
• Since kinetic energy is directly influenced by the velocity of the particle, any increase or decrease in its speed will change its kinetic energy.

With this information, we can connect it to other forms of energy like potential energy, contributing to the overall mechanical energy of the system.

Mechanical Energy

Mechanical energy is the sum of potential and kinetic energy in a system. It represents the total energy available for movement and is a key concept in understanding particle dynamics.
The equation for mechanical energy (ME) is: \[ ME = KE + U \] Using this formula, we can find the mechanical energy at x = 6.0 m:

• Given that the kinetic energy (KE) at x = 6.0 m is 20 J,
• and potential energy (U) is 14 J,
• we substitute these values into the equation: \[ ME = 20 + 14 = 34 \, \text{J} \]

Thus, the mechanical energy of the system at this point is 34 Joules, which corresponds to option (A) in the given choices.

Particle Dynamics

Particle dynamics involves understanding and predicting the motion of particles as they respond to forces and energy changes. It combines principles like Newton's Laws of Motion and concepts of energy conservation.
In the exercise, the particle moves under a conservative force, where energy exchanges between potential and kinetic energy without any loss to the surroundings.

• Conservative forces, such as the one acting on the particle, ensure total mechanical energy is preserved.
• The particle’s path is along the x-axis, and because of conservation laws, mechanical energy can be used to predict future motion or verify current motion is correct.

By understanding particle dynamics, we can assess movements and predict how forces and energy will affect particle trajectories.