Calculating the Maximum Height Reached by a Block Launched from a Spring

In this article, we will explore the physics behind launching a block from a compressed spring. We will use the principle of conservation of energy to calculate the maximum height that the block reaches when it is launched. This process involves understanding the potential energy stored in the spring, converting it to gravitational potential energy, and solving for the height reached by the block.

Overview of the Problem

A block with a mass of 785 grams (0.785 kg) is placed on a spring with a spring constant of 105 N/m. The spring is compressed by 33 cm (0.33 m) from its equilibrium position. We need to calculate the height the block reaches when the spring is released.

Step-by-Step Solution

Step 1: Calculate the Potential Energy Stored in the Spring

The potential energy stored in a compressed spring can be calculated using the formula:

P[E_{_{text{spring}}}] 12 k x^2

where:

k is the spring constant (105 N/m) x is the compression of the spring in meters (0.33 m)

First, we convert the compression from centimeters to meters:

x 33 cm 0.33 m

Next, we plug in the values:

P[E_{_{text{spring}}}] 12 × 105 N/m × (0.33 m)^2

Calculating this:

P[E_{_{text{spring}}}]≈ 12 × 11.4295 J

P[E_{_{text{spring}}}]≈ 5.71475 J

Step 2: Calculate the Gravitational Potential Energy at Maximum Height

The gravitational potential energy at the maximum height can be calculated using the formula:

P[E_{_{text{gravity}}}] mgh

where:

m is the mass of the block (0.785 kg) g is the acceleration due to gravity (9.81 m/s2) h is the height reached

Setting the potential energy from the spring equal to the gravitational potential energy:

P[E_{_{text{spring}}}] P[E_{_{text{gravity}}}]

5.71475 J 0.785 kg × 9.81 m/s2 × h

Step 3: Solve for h

Rearranging the equation to solve for h:

h 5.71475 J / (0.785 kg × 9.81 m/s2)

Calculating this:

h 5.71475 / 7.69285 ≈ 0.741 m

Therefore, the height that the block reaches when it is launched is approximately 0.741 meters (or 74.1 cm).

Conclusion

By using the principle of conservation of energy, we can accurately calculate the height that the block reaches after being launched from a compressed spring. The block reaches a height of approximately 0.741 meters above its initial position, which is approximately 0.48 meters above the equilibrium position of the spring.

Further Discussion

It is important to note that the spring constant, gravitational acceleration, and mass of the block play crucial roles in determining the final height. In this scenario, we have assumed negligible friction, and the spring has separated from the block at the maximum height. This result allows us to understand the energy transfer and the dynamics involved in such a mechanical system.