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Elio Fabri
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Suppose a spring is on a surface.An An object of mass $m$ is in $h$ meter height from the top of the spring.When When the object falls freely,the the spring contracts by $x$ meter.Find Find $x$ when $k$ is spring constant.

The solution given in books is $mg(h+x)=\frac{1}{2}kx^2$.

Now,let us consider the top of the spring as reference plane.So So potential energy at that plane is $0$.Now Now energy of the system at the beginning is only the potential energy of the object.So

So $E_1=mgh$

After the fall,the spring contracts by $x$ meter,so so the potential energy of the spring is now $\frac{1}{2}kx^2$ and the potential energy of the object is $-mgx$ and kinetic energy of the object is   

$\frac{1}{2}mv^2$.

So by conservation of energy,we get

$mgh=-mgx+\frac{1}{2}mv^2+\frac{1}{2}kx^2$.

Where did I go wrong?

Suppose a spring is on a surface.An object of mass $m$ is in $h$ meter height from the top of the spring.When the object falls freely,the spring contracts by $x$ meter.Find $x$ when $k$ is spring constant.

The solution given in books is $mg(h+x)=\frac{1}{2}kx^2$.

Now,let us consider the top of the spring as reference plane.So potential energy at that plane is $0$.Now energy of the system at the beginning is only the potential energy of the object.So

$E_1=mgh$

After the fall,the spring contracts by $x$ meter,so the potential energy of the spring is now $\frac{1}{2}kx^2$ and the potential energy of the object is $-mgx$ and kinetic energy of the object is  $\frac{1}{2}mv^2$.

So by conservation of energy,we get

$mgh=-mgx+\frac{1}{2}mv^2+\frac{1}{2}kx^2$.

Where did I go wrong?

Suppose a spring is on a surface. An object of mass $m$ is in $h$ meter height from the top of the spring. When the object falls freely, the spring contracts by $x$ meter. Find $x$ when $k$ is spring constant.

The solution given in books is $mg(h+x)=\frac{1}{2}kx^2$.

Now,let us consider the top of the spring as reference plane. So potential energy at that plane is $0$. Now energy of the system at the beginning is only the potential energy of the object. So $E_1=mgh$

After the fall,the spring contracts by $x$ meter, so the potential energy of the spring is now $\frac{1}{2}kx^2$ and the potential energy of the object is $-mgx$ and kinetic energy of the object is 

$\frac{1}{2}mv^2$.

So by conservation of energy,we get

$mgh=-mgx+\frac{1}{2}mv^2+\frac{1}{2}kx^2$.

Where did I go wrong?

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a_i_r
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Conservation of energy on nail problems

Suppose a spring is on a surface.An object of mass $m$ is in $h$ meter height from the top of the spring.When the object falls freely,the spring contracts by $x$ meter.Find $x$ when $k$ is spring constant.

The solution given in books is $mg(h+x)=\frac{1}{2}kx^2$.

Now,let us consider the top of the spring as reference plane.So potential energy at that plane is $0$.Now energy of the system at the beginning is only the potential energy of the object.So

$E_1=mgh$

After the fall,the spring contracts by $x$ meter,so the potential energy of the spring is now $\frac{1}{2}kx^2$ and the potential energy of the object is $-mgx$ and kinetic energy of the object is $\frac{1}{2}mv^2$.

So by conservation of energy,we get

$mgh=-mgx+\frac{1}{2}mv^2+\frac{1}{2}kx^2$.

Where did I go wrong?