A 1.2kg block is dropped from 48cm above a spring in equilibrium. The force constant for the spring is 124N/m. Calculate the maximum compression in the spring.
If X is maximum spring compression, and H is the height from which the block falls (0.48 m), conservation of energy tell you that
M g (H + X) = (1/2) k X^2
where k is the spring's force constant.
Solve for X
Take the positive root of the resulting quadratic equation.
To calculate the maximum compression in the spring, we can use the principle of conservation of mechanical energy.
The potential energy of the block when it is dropped is given by the formula:
Potential energy (PE) = m * g * h
where m is the mass of the block (1.2 kg), g is the acceleration due to gravity (9.8 m/s²), and h is the height from which the block is dropped (48 cm).
PE = 1.2 kg * 9.8 m/s² * 0.48 m
PE = 5.568 J
At the maximum compression, all of the potential energy of the block will be stored as the elastic potential energy of the spring.
The elastic potential energy (PE_spring) of a spring is given by the formula:
PE_spring = (1/2) * k * x²
where k is the force constant of the spring (124 N/m) and x is the compression of the spring.
Since the potential energy of the block is equal to the elastic potential energy of the spring, we can write:
PE = PE_spring
5.568 J = (1/2) * 124 N/m * x²
Rearranging the equation to solve for x:
x² = (2 * 5.568 J) / 124 N/m
x² = 0.089
Taking the square root of both sides:
x = √0.089
x ≈ 0.298 m
Therefore, the maximum compression in the spring is approximately 0.298 meters.
To calculate the maximum compression in the spring, we need to first find the gravitational potential energy of the block when it is dropped from a height of 48 cm. Then, we can equate this energy to the potential energy stored in the compressed spring.
1. Find the gravitational potential energy (Ug) of the block:
The gravitational potential energy can be calculated using the formula:
Ug = m * g * h
where m is the mass of the block, g is the acceleration due to gravity, and h is the height from which the block is dropped.
Given:
Mass of the block (m) = 1.2 kg
Height (h) = 48 cm = 0.48 m
Acceleration due to gravity (g) = 9.8 m/s^2
Substituting the values into the formula:
Ug = 1.2 kg * 9.8 m/s^2 * 0.48 m
Ug ≈ 5.628 J
2. Equate the gravitational potential energy to the potential energy stored in the compressed spring:
The potential energy stored in a spring (Us) can be calculated using the formula:
Us = (1/2) * k * x^2
where k is the force constant of the spring and x is the maximum compression of the spring.
Given:
Force constant of the spring (k) = 124 N/m
Substituting the values into the formula:
5.628 J = (1/2) * 124 N/m * x^2
Rearranging the equation to solve for x:
x^2 = (2 * 5.628 J) / (124 N/m)
x^2 ≈ 0.0909 m^2
Taking the square root of both sides:
x ≈ √(0.0909 m^2)
x ≈ 0.3015 m
Therefore, the maximum compression in the spring is approximately 0.3015 meters.