Problem:
A 2g block attached to an ideal spring with a spring constant of 80N/m oscillates on a horizontal frictionless surface. When the spring is 4.0 cm extended beyond its equilibrium length, the speed of the block is √17 m/s. What is the greatest speed of the block?

Solution:

Given:
- Mass of the block (m) = 2g = 0.002 kg
- Spring constant (k) = 80 N/m
- Extension of the spring (x) = 4.0 cm = 0.04 m
- Speed of the block at x = √17 m/s

Analysis:
To find the greatest speed of the block, we need to determine the maximum amplitude of oscillation. The block oscillates around the equilibrium position due to the force provided by the spring. At maximum amplitude, the block momentarily comes to rest before reversing its direction. This point is called the turning point.

Procedure:
1. Find the total mechanical energy of the block-spring system.
2. Use the total mechanical energy to find the maximum amplitude.
3. Calculate the greatest speed of the block using the maximum amplitude.

Step 1: Find the total mechanical energy:
The total mechanical energy (E) of the block-spring system is the sum of its kinetic energy (KE) and potential energy (PE) at any point during the oscillation.

E = KE + PE

At the maximum amplitude (turning point), the block momentarily comes to rest, so its kinetic energy is zero.

E = PE

Potential energy stored in the spring (PE) can be calculated using the equation:

PE = (1/2) k x^2

Substituting the given values:

PE = (1/2) * 80 N/m * (0.04 m)^2
PE = 0.064 J

Therefore, the total mechanical energy at the turning point is 0.064 J.

Step 2: Find the maximum amplitude:
At the turning point, the total mechanical energy is equal to the potential energy.

E = PE = (1/2) k x^2

Rearranging the equation:

x^2 = (2 * PE) / k
x^2 = (2 * 0.064 J) / 80 N/m
x^2 = 0.0016 m^2

Taking the square root of both sides:

x = 0.04 m

Therefore, the maximum amplitude (A) is 0.04 m.

Step 3: Calculate the greatest speed:
The greatest speed of the block occurs at the equilibrium position (x = 0). At this point, all the potential energy is converted into kinetic energy.

E = KE

Using the total mechanical energy equation:

E = (1/2) k x^2

Substituting the given values:

E = (1/2) * 80 N/m * (0.04 m)^2
E = 0.064 J

Therefore, the kinetic energy (KE) at the equilibrium position is 0.064 J.

The kinetic energy can be calculated using the equation:

KE = (1/2) m v^2

Substituting the known