A 2.50-kg mass is pushed against a horizontal spring of force constant 25.0 N/cm on a frictionless air table. The spring is attached to the tabletop, and the mass is not attached to the spring in any way. When the spring has been compressed enough to store 11.5 J of potential energy in it, the mass is suddenly released from rest. (a) Find the greatest speed the mass reaches. When does this occur? (b) What is the greatest acceleration of the mass, and when does it occur?

Given Data:

The force constant of the spring is k = 25 N/cm

The mass of the object is: m = 2.50 kg .

The potential energy for spring compression is U = 11.5 J

The mass of the book is m = 1.20 m

The height of the book is h = 0.800 m

Potential Energy:

The energy stored due to the variation in the position with some reference position is the potential energy. The potential energy is used to calculate the compression of the spring.

(a)

The greatest speed of mass is calculated as:

$$\begin{gathered} \mathrm{U}=\frac{1}{2}{\mathrm{mv}}^2 \\ \mathrm{v}=\sqrt{\frac{2\mathrm{U}}{\mathrm{m}}} \end{gathered}$$

Here, U the potential energy of spring m is the mass of the object.

Substitute all the values in the above equation and we get,

$$\begin{gathered} \mathrm{v}=\sqrt{\frac{2\left(11.5\mathrm{J}\right)}{2.50\mathrm{kg}}} \\ \mathrm{v}=3.03\mathrm{m}/\mathrm{s} \end{gathered}$$

Therefore, the greatest speed of mass is 3.03 m/s .

(b)

The maximum compressed distance is calculated as:

$\mathrm{U}=\frac{1}{2}{\mathrm{kx}}^2$

Here, k is force constant and x is compression of spring.

Substitute all the values in the above equation, and we get,

$$\begin{gathered} 11.5=\frac{1}{2}\left(25\mathrm{N}/\mathrm{cm}\right)\left(\frac{100\mathrm{cm}}{1\mathrm{m}}\right){\mathrm{x}}^2 \\ \mathrm{x}=0.096\mathrm{m} \end{gathered}$$

The greatest acceleration of the mass is calculated as:

ma = kx

Substitute all the values in the above equation.

$$\begin{gathered} \left(2.50\mathrm{kg}\right)\times \mathrm{a}\left(25\mathrm{N}/\mathrm{cm}\right)\left(\frac{100\mathrm{cm}}{1\mathrm{m}}\right)\left(0.096\right) \\ \mathrm{a}=96\mathrm{m}/{\mathrm{s}}^2 \end{gathered}$$

Therefore, the maximum acceleration of the mass is $96\mathrm{m}/{\mathrm{s}}^2$.