Question

An object of mass m attached to a spring of force constant k oscillates with simple harmonic motion. The maximum displacement from equilibrium is A and the total mechanical energy of the system is E.What is the object's velocity when its potential energy is 2/3E?

Answer 1

As we know that total energy of SHM is always constant

so here we know that

total energy = Kinetic energy + potential energy

so here we have

potential energy = 2/3E

so we will have kinetic energy as

$KE = E - \frac{2}{3}E$

now we have

$KE = \frac{1}{3}E$

now for the speed of the block

$\frac{1}{2} mv^2 = \frac{1}{3}E$

by solving above equation we have

$v = \sqrt{\frac{2E}{3m}}$

Answer 2

The object's velocity is √[(kA²)/(3m)]

$\texttt{ }$

Further explanation

Hooke's Law states that the length of a spring is directly proportional to the force acting on the spring.

$\boxed {F = k \times \Delta x}$

F = Force ( N )

k = Spring Constant ( N/m )

Δx = Extension ( m )

$\texttt{ }$

The formula for finding Young's Modulus is as follows:

$\boxed {E = \frac{F / A}{\Delta x / x_o}}$

E = Young's Modulus ( N/m² )

F = Force ( N )

A = Cross-Sectional Area ( m² )

Δx = Extension ( m )

x = Initial Length ( m )

Let us now tackle the problem !

$\texttt{ }$

Given:

mass of the object = m

force constant = k

maximum displacement = A

total mechanical energy = E

potential energy = ²/₃E

Unknown:

velocity of the object = v = ?

Solution:

We will use conservation of energy as follows:

$\texttt{Total Mechanical Energy} = E_p + E_k$

$E = \frac{2}{3}E + \frac{1}{2}mv^2$

$E - \frac{2}{3}E = \frac{1}{2}mv^2$

$\frac{1}{3}E = \frac{1}{2}mv^2$

$\frac{1}{3} (\frac{1}{2}kA^2) = \frac{1}{2}mv^2$

$\frac{1}{3}kA^2 = mv^2$

$v^2 = \frac{1}{3}kA^2 \div m$

$\large {\boxed {v = \sqrt{ \frac{kA^2}{3m}}}}$

$\texttt{ }$

Learn more

• Young's modulus : brainly.com/question/6864866
• Young's modulus for aluminum : brainly.com/question/7282579
• Young's modulus of wire : brainly.com/question/9755626

$\texttt{ }$

Answer details

Grade: College

Subject: Physics

Chapter: Elasticity

$\texttt{ }$

Keywords: Elasticity , Diameter , Concrete , Column , Load , Compressed , Stretched , Modulus , Young