Show that if we increase the k?netic energy of a system without decreasing the potential energy (for example, we increase the mass on a given spring), then every characteristic frequency decreases.

Short Answer

Expert verified
Answer: When the kinetic energy of a spring-mass system is increased without decreasing its potential energy, for instance by increasing the mass on the spring, every characteristic frequency of the system decreases.

Step by step solution

01

Recall the equations for kinetic and potential energy

Kinetic energy is given by the equation K = (1/2)mv^2, where m is the mass of the object and v is its velocity. The potential energy of a spring-mass system is given by U = (1/2)kx^2, where k is the spring constant and x is the displacement from the equilibrium position.
02

Define the characteristic frequency of the system

The characteristic frequency, f, of a spring-mass system is given by the formula f = (1/2π)√(k/m), where k is the spring constant and m is the mass attached to the spring.
03

Set the conditions for our problem

We are told that we increase the kinetic energy of the system without decreasing its potential energy. This means that the mass, m, on the spring is increased while keeping the potential energy, U, constant.
04

Express the spring constant as a function of the potential energy

From the potential energy equation, U = (1/2)kx^2, we can express the spring constant, k, as a function of the potential energy: k = 2U/x^2.
05

Substituting the spring constant in the characteristic frequency formula

Replace the spring constant, k, in the expression for characteristic frequency, f, with the expression we found in step 4: f = (1/2π)√((2U/x^2)/m).
06

Demonstrate that an increase in mass leads to a decrease in characteristic frequency

Notice that in the expression for f, the mass m is in the denominator. So, when m increases (while keeping U and x constant), the entire expression (2U/x^2)/m will decrease. Since the square root of a smaller number is also smaller, the characteristic frequency f will decrease as a result. In conclusion, when the kinetic energy of a system is increased without decreasing its potential energy, for instance by increasing the mass on the spring, every characteristic frequency of the system decreases.

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