A 4.30 kg block starts from rest and slides down a frictionless incline, dropping a vertical distance of 2.60 m, before compressing a spring of force constant 2.20 104 N/m. Find the maximum compression of the spring.
change of PE= change of KE
mg*2.60= 1/2 k x^2 solve for x.
To find the maximum compression of the spring, we need to calculate the potential energy lost by the block as it slides down the incline, and then equate it to the potential energy stored in the compressed spring.
First, let's calculate the potential energy lost by the block as it slides down the incline. The formula for potential energy is given as:
Potential energy (PE) = mass (m) × gravitational acceleration (g) × height (h)
Given that the mass (m) of the block is 4.30 kg, the gravitational acceleration (g) is approximately 9.8 m/s², and the height (h) is 2.60 m, we can substitute these values into the formula to calculate the potential energy lost:
PE = 4.30 kg × 9.8 m/s² × 2.60 m
PE = 108.044 J
Now that we know the potential energy lost by the block, we can equate it to the potential energy stored in the compressed spring. The formula for potential energy stored in a spring is given as:
Potential energy (PE) = 0.5 × spring constant (k) × compression²
We can rearrange this formula to solve for the compression:
Compression² = (2 × PE) / k
Compression² = (2 × 108.044 J) / (2.20 × 10⁴ N/m)
By substituting the respective values into the formula, we can calculate the compression:
Compression² ≈ 0.0098 m²
Taking the square root of both sides, we find:
Compression ≈ 0.099 m or 9.9 cm
Therefore, the maximum compression of the spring is approximately 0.099 m or 9.9 cm.