A block of mass m = 0.72 kg attached to a spring with force constant 137 N/m is free to move on a frictionless, horizontal surface as in the figure below. The block is released from rest after the spring is stretched a distance A = 0.13 m. (Indicate the direction with the sign of your answer. Assume that the positive direction is to the right.)

The left end of a horizontal spring is attached to a vertical wall, and the right end is attached to a block of mass m which can slide left and right on a horizontal surface. The spring is stretched from its equilibrium position by a distance A.
(a) At that instant, find the force on the block.

(b) At that instant, find its acceleration.

To solve this problem, we will use Hooke's Law and Newton's Second Law of Motion.

### (a) Force on the block

The force exerted by a spring is given by Hooke's Law:

\[ F = -k \cdot x \]

where:
- \( F \) is the force exerted by the spring,
- \( k \) is the spring constant, and
- \( x \) is the displacement from the equilibrium position.

Given:
- \( k = 137 \, \text{N/m} \)
- \( x = A = 0.13 \, \text{m} \)

Substitute the values:

\[ F = -137 \, \text{N/m} \times 0.13 \, \text{m} \]

The negative sign indicates that the force is in the opposite direction of the displacement, i.e., towards the left when the spring is stretched.

### (b) Acceleration of the block

The acceleration can be found using Newton's Second Law:

\[ F = m \cdot a \]

where:
- \( F \) is the force on the block,
- \( m \) is the mass of the block, and
- \( a \) is the acceleration of the block.

We already have \( F \) from part (a) and \( m = 0.72 \, \text{kg} \). We can solve for \( a \):

\[ a = \frac{F}{m} \]

Let's calculate the force and acceleration.(a) The force on the block is \( -17.81 \, \text{N} \). The negative sign indicates that the force is directed to the left.

(b) The acceleration of the block is \( -24.74 \, \text{m/s}^2 \). The negative sign indicates that the acceleration is also directed to the left.

Would you like more details or have any questions?

**Related Questions:**
1. What would be the maximum velocity of the block as it oscillates?
2. How much work is done by the spring force in one complete oscillation?
3. How would the force change if the spring constant were doubled?
4. What is the total mechanical energy of the system?
5. How would the motion be affected if the surface had friction?
6. What is the frequency of oscillation for this system?
7. How does the displacement affect the acceleration of the block?
8. What is the potential energy stored in the spring when stretched to 0.13 m?

**Tip:** When dealing with simple harmonic motion, the acceleration and force are always directed toward the equilibrium position, which is opposite the direction of displacement.

Learn how to calculate the force and acceleration of a block attached to a spring using Hooke's Law and Newton's Second Law of Motion. This problem involves a horizontal spring attached to a block of mass 0.72 kg, stretched by a distance of 0.13 m. Understand the physics behind force, acceleration, and springs in this advanced high school level problem.

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