Spring is stretched by force f to distance "x"
now here by force balance we can say
[tex]f = kx[/tex]
[tex]k = \frac{f}{x}[/tex]
now here we will we say that energy stored in the spring will convert into kinetic energy
[tex]\frac{1}{2} kx^2 = \frac{1}{2}mv^2[/tex]
[tex]\frac{f}{x} (x^2} = mv^2[/tex]
now solving above equation we will have
[tex]v =\sqrt{ \frac{fx}{m}}[/tex]
PART 2)
now for half of the extension again we can use energy conservation
[tex]\frac{1}{2}kx^2 - \frac{1}{2}k(x/2)^2 = \frac{1}{2} mv^2[/tex]
[tex]\frac{3}{4}kx^2 = mv^2[/tex]
[tex]\frac{3}{4}fx = mv^2[/tex]
now the speed is given as
[tex]v = \sqrt{\frac{3fx}{4m}}[/tex]
The speed of the mass m when the spring returns:
(a) to its normal length ( x = 0 ) → v = √ ( F x / m )
(b) to half its original extension ( x/2 ) → v = √ ( 3F x / 4m )
[tex]\texttt{ }[/tex]
Let's recall Elastic Potential Energy formula as follows:
[tex]\boxed{E_p = \frac{1}{2}k x^2}[/tex]
where:
Ep = elastic potential energy ( J )
k = spring constant ( N/m )
x = spring extension ( compression ) ( m )
Let us now tackle the problem!
[tex]\texttt{ }[/tex]
Given:
initial extension of the spring = x
force = F
mass of the object = m
Asked:
speed of the mass = v = ?
Solution:
We will use Conservation of Energy formula to solve this problem.
[tex]Ep_1 + Ek_1 = Ep_2 + Ek_2[/tex]
[tex]\frac{1}{2}k x^2 + 0 = 0 + \frac{1}{2}m v^2[/tex]
[tex]\frac{1}{2}k x^2 = \frac{1}{2}m v^2[/tex]
[tex]k x^2 = m v^2[/tex]
[tex]v^2 = k x^2 \div m[/tex]
[tex]v = \sqrt{ \frac {k x^2}{ m } }[/tex]
[tex]\boxed {v = \sqrt{ \frac {F x}{ m }} }[/tex]
[tex]\texttt{ }[/tex]
[tex]Ep_1 + Ek_1 = Ep_2 + Ek_2[/tex]
[tex]\frac{1}{2}k x^2 + 0 = \frac{1}{2}k (x/2)^2 + \frac{1}{2}m v^2[/tex]
[tex]\frac{1}{2}k x^2 - \frac{1}{2}k (x/2)^2 = \frac{1}{2}m v^2[/tex]
[tex]\frac{3}{4}k x^2 = m v^2[/tex]
[tex]v^2 = \frac{3}{4} k x^2 \div m[/tex]
[tex]v = \sqrt{ \frac {3k x^2}{ 4m } }[/tex]
[tex]\boxed {v = \sqrt{ \frac {3F x}{ 4m } }}[/tex]
[tex]\texttt{ }[/tex]
[tex]\texttt{ }[/tex]
Grade: High School
Subject: Physics
Chapter: Elasticity
