2. The potential energy function associated with a force acting on a system is U = 3x3 – 5x. a) What is the equation for force for this system? b) Draw a graph of the potential energy. Indicate the points of equilibrium and note whether they are stable, unstable or neutral.
Kinematics
A machine is a device that accepts energy in some available form and utilizes it to do a type of work. Energy, work, or power has to be transferred from one mechanical part to another to run a machine. While the transfer of energy between two machine parts, those two parts experience a relative motion with each other. Studying such relative motions is termed kinematics.
Kinetic Energy and Work-Energy Theorem
In physics, work is the product of the net force in direction of the displacement and the magnitude of this displacement or it can also be defined as the energy transfer of an object when it is moved for a distance due to the forces acting on it in the direction of displacement and perpendicular to the displacement which is called the normal force. Energy is the capacity of any object doing work. The SI unit of work is joule and energy is Joule. This principle follows the second law of Newton's law of motion where the net force causes the acceleration of an object. The force of gravity which is downward force and the normal force acting on an object which is perpendicular to the object are equal in magnitude but opposite to the direction, so while determining the net force, these two components cancel out. The net force is the horizontal component of the force and in our explanation, we consider everything as frictionless surface since friction should also be calculated while called the work-energy component of the object. The two most basics of energy classification are potential energy and kinetic energy. There are various kinds of kinetic energy like chemical, mechanical, thermal, nuclear, electrical, radiant energy, and so on. The work is done when there is a change in energy and it mainly depends on the application of force and movement of the object. Let us say how much work is needed to lift a 5kg ball 5m high. Work is mathematically represented as Force ×Displacement. So it will be 5kg times the gravitational constant on earth and the distance moved by the object. Wnet=Fnet times Displacement.
![**Problem 2: Analysis of Potential Energy and Force**
The potential energy function associated with a force acting on a system is given by:
\[ U = 3x^3 - 5x. \]
**Questions:**
a) What is the equation for the force for this system?
b) Draw a graph of the potential energy. Indicate the points of equilibrium and note whether they are stable, unstable, or neutral.
**Solutions:**
a) **Equation for Force:**
To find the force exerted by the system, we use the relation between force and potential energy:
\[ F(x) = -\frac{dU}{dx}. \]
Calculate the derivative of the potential energy function:
\[ U = 3x^3 - 5x, \]
\[ \frac{dU}{dx} = 9x^2 - 5. \]
Hence, the force is:
\[ F(x) = -(9x^2 - 5) = -9x^2 + 5. \]
b) **Graph of Potential Energy:**
To analyze the points of equilibrium, we find where the derivative of \( U \) (or equivalently, where \( F(x) = 0 \)) is zero. This gives equilibrium points:
\[ 9x^2 - 5 = 0, \]
\[ 9x^2 = 5, \]
\[ x^2 = \frac{5}{9}, \]
\[ x = \pm\frac{\sqrt{5}}{3}. \]
To determine the stability of these points, evaluate the second derivative:
\[ \frac{d^2U}{dx^2} = 18x. \]
- At \( x = \frac{\sqrt{5}}{3} \), \( \frac{d^2U}{dx^2} = 18\left(\frac{\sqrt{5}}{3}\right) > 0 \). This indicates a local minimum: **Stable Equilibrium.**
- At \( x = -\frac{\sqrt{5}}{3} \), \( \frac{d^2U}{dx^2} = 18\left(-\frac{\sqrt{5}}{3}\right) < 0 \). This indicates a local maximum: **Unstable Equilibrium.**
**Diagram Explanation:**
- Plot the curve of \( U = 3x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47172a01-dc9c-43a5-97e6-7fd3dbf42a8c%2Fa9563b7b-fc2a-4ad7-bfe0-a22866625fe6%2F8kh397r_processed.png&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 5 steps with 1 images
