Answer The Following Questions In Your Own Words Describe Th

Answer The Following Questionsn Your Own Words Describe The P

Answer the following questions:

- In your own words, describe the process of energy exchange with the spring (Without air resistance).

- When we add air resistance, is the rate at which energy is dissipated constant? (Think of drag, is it constant with velocity?)

- Draw two graphs which include kinetic energy, potential energy of the spring, and total mechanical energy. The first graph is without air resistance, the second one with air resistance.

Answer the true or false questions:

- The function that describes potential energy increases/decreases linearly. – False

- The work done on an object depends on the force and the distance the object travels. – True

- Total mechanical energy increases with time in an isolated system, just like entropy. – False

- Friction force changes depending on the force applied to the object. – True

- According to Hooke’s law, the force applied by a spring is opposite to the displacement. – True

Calculate the work done by friction on Kenny:

Kenny, with a mass of 30 kg, slides from a height of 1000 meters, experiences friction, and exits the slide at ground level with a velocity of 100 m/s, which causes his death. Determine the work done by friction. Is the sign of this value positive or negative? Explain why.

A spring with a 3 kg mass attached hangs vertically, stretching 1 centimeter due to the weight. Find the spring constant.

Paper For Above instruction

The exploration of energy exchange in spring systems and the impact of dissipative forces such as air resistance and friction is fundamental in understanding classical mechanics. This essay discusses the energy transfer processes in ideal and real systems, visualizes energy variations through graphs, evaluates empirical statements about energy and forces, and applies these principles to practical problems involving work and spring constants.

Energy exchange with a spring (without air resistance)

In an ideal scenario absent of air resistance, the energy exchange in a spring system can be described by the conservation of mechanical energy. When a spring is compressed or stretched from its equilibrium position, it stores potential energy proportional to the displacement, according to Hooke's law. Specifically, the potential energy stored in a spring is given by \(PE_s = \frac{1}{2} k x^2\), where \(k\) is the spring constant and \(x\) is the displacement. As the system oscillates, energy converts between potential energy in the spring and kinetic energy, \(KE = \frac{1}{2} m v^2\), with total mechanical energy remaining constant. When the mass moves towards maximum displacement, kinetic energy diminishes to zero, and potential energy peaks; conversely, at the equilibrium point, potential energy is zero and kinetic energy is at its maximum. This cyclic transfer illustrates an energy exchange process that theoretically conserves total energy in an ideal, frictionless environment.

Energy dissipation when air resistance is included

Introducing air resistance, mainly modeled as drag force, results in energy dissipation from the system. Unlike ideal cases, the rate at which energy dissipates due to drag is not constant. Air resistance is generally proportional to velocity (or its square), depending on the regime. For example, viscous drag in laminar flow is proportional to velocity, \(F_d = -b v\), where \(b\) is a damping coefficient, leading to a decelerating force that depends on current velocity. As velocity increases, the drag force grows, meaning energy is dissipated more rapidly at higher speeds. Consequently, the rate of energy loss is not constant but varies with velocity, typically increasing as velocity increases, and decreasing as the system slows down. This non-uniform energy dissipation results in a gradual reduction of total mechanical energy over time.

Graphical analysis of energy in spring systems

Graph 1: Without air resistance, the total mechanical energy remains constant. Kinetic energy and potential energy of the spring oscillate inversely; when kinetic energy is at a maximum, potential energy is zero, and vice versa. The total energy line remains flat, illustrating conservation.

Graph 2: With air resistance, the total mechanical energy decreases over time. The oscillations of kinetic and potential energy diminish in amplitude as energy is dissipated. The total energy graph shows a declining trend, asymptotically approaching zero, reflecting continuous energy loss due to drag forces.

Assessment of statements about potential energy, work, energy and forces

• Potential energy increases/decreases linearly: False — For springs, potential energy varies quadratically with displacement, not linearly.
• The work done depends on force and distance traveled: True — Work is defined as the integral of force over displacement, thus depending on force magnitude and path length.
• Total mechanical energy increases with time in an isolated system: False — In an ideal isolated system, energy remains constant; with dissipative forces, it decreases over time.
• Friction force varies depending on the applied force: False — Static friction varies with applied force up to a maximum; kinetic friction is generally constant independent of applied force (assuming surfaces and conditions are steady).
• Spring force opposes displacement: True — According to Hooke's law, the restoring force is opposite to the displacement direction.

Work done by friction on Kenny

Kenny's initial potential energy at height \(h = 1000\,m\): \(PE = mgh = 30 \times 9.81 \times 1000 = 294,300\,J\). His final kinetic energy upon exiting the slide: \(KE = \frac{1}{2} \times 30 \times (100)^2 = 150,000\,J\). The work-energy principle states that initial potential energy minus final kinetic energy equals the work done by non-conservative forces (i.e., friction):

\[W_{friction} = PE_{initial} - KE_{final} = 294,300\,J - 150,000\,J = 144,300\,J\]

Since energy is lost due to friction, the work done by friction is negative from the system’s perspective, representing energy dissipated as heat or sound. The magnitude is 144,300 J, and the negative sign indicates energy removal from the system.

Calculating spring constant

The spring stretches 1 centimeter (\(x = 0.01\,m\)) under the weight of the 3 kg mass. The force due to the weight is \(F = mg = 3 \times 9.81 = 29.43\,N\). According to Hooke’s law:

\[F = kx \Rightarrow k = \frac{F}{x} = \frac{29.43}{0.01} = 2943\,N/m\]

Thus, the spring constant is approximately 2943 N/m.

Conclusion

The analysis highlights fundamental principles of energy conservation, dissipation, and elastic forces. Understanding how energy exchanges differently in ideal versus real systems provides insights into the behavior of physical systems encountered in engineering and physics. The ability to quantify energy loss through work calculations and determine spring constants forms the basis for designing systems that optimize energy efficiency and mechanical stability.

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