An 8.46 kg block drops straight down from a height of 0.90 m, striking a platform spring having a force constant of 9.90 102 N/m. Find the maximum compression of the spring

To find the maximum compression of the spring, we can use the concept of conservation of energy.

When the block falls from a height of 0.90 m, it gains potential energy which gets converted into the elastic potential energy stored in the spring when it compresses.

The potential energy gained by the block can be calculated using the formula:

Potential Energy (PE) = m * g * h

Where:
m = mass of the block (8.46 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height from which the block falls (0.90 m)

PE = 8.46 kg * 9.8 m/s^2 * 0.90 m
PE = 76.5672 J (rounded to four decimal places)

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

Elastic Potential Energy (PE) = (1/2) * k * x^2

Where:
k = force constant of the spring (9.90 * 10^2 N/m)
x = maximum compression of the spring (what we need to find)

We can equate the potential energy gained by the block to the elastic potential energy stored in the spring:

PE = (1/2) * k * x^2
76.5672 J = (1/2) * 9.90 * 10^2 N/m * x^2

Let's solve for x:

x^2 = (2 * 76.5672 J) / (9.90 * 10^2 N/m)
x^2 = 1.543568686 (rounded to nine decimal places)

Taking the square root of both sides:

x ≈ √1.543568686
x ≈ 1.24 m (rounded to two decimal places)

Therefore, the maximum compression of the spring is approximately 1.24 meters.