a block with a mass of 5.00 kg is moving at 8.00 m/s^-1 along a frictionless horizonal surface toward a spring with force constant k= 500 N/m^1 that is attached to a wall.find the maximum distance the spring is compressed. the spring has negligible mass.
To find the maximum distance the spring is compressed, we can use the concept of conservation of mechanical energy.
Given:
Mass of the block, m = 5.00 kg
Initial velocity of the block, u = 8.00 m/s
Force constant of the spring, k = 500 N/m
First, let's find the mechanical energy of the block.
The mechanical energy of the block can be calculated as the sum of its kinetic energy and potential energy. Since the block is initially moving, it has kinetic energy and no potential energy.
The kinetic energy (KE) of the block is given by the formula:
KE = (1/2) * m * u^2
Substituting the given values, we have:
KE = (1/2) * 5.00 kg * (8.00 m/s)^2
= 160 J
At maximum compression, the block momentarily comes to rest. At this point, all the kinetic energy is converted into potential energy stored in the spring.
The potential energy (PE) stored in the spring can be calculated using Hooke's Law, which states that the potential energy stored in a spring is directly proportional to the square of the displacement from its equilibrium position.
The potential energy (PE) of the spring is given by the formula:
PE = (1/2) * k * x^2
Where:
k = Force constant of the spring = 500 N/m
x = Maximum compression of the spring (unknown)
Now equating the kinetic energy and potential energy, we have:
KE = PE
(1/2) * m * u^2 = (1/2) * k * x^2
Substituting the given values, we have:
(1/2) * 5.00 kg * (8.00 m/s)^2 = (1/2) * (500 N/m) * x^2
Simplifying the equation:
160 J = 250 N/m * x^2
x^2 = (160 J) / (250 N/m)
x^2 ≈ 0.64 m^2
To find the maximum distance the spring is compressed, we take the square root of both sides:
x ≈ √(0.64 m^2)
x ≈ 0.8 m
Therefore, the maximum distance the spring is compressed is approximately 0.8 meters.