A 200 g mass is attached to a spring of spring constant k. The spring is compressed 15 cm from its equilibrium value. When released the mass reaches a speed of 5 m/s. What is the spring constant (in N/m)?
w*(amplitude) = maximum velocity
Solve for w (in radians/second) and then use
w = sqrt(k/m) to get k, the spring constant
Amplitudes and velocities must be in meters and m/s. Mass must be in kg
0.25m
To find the spring constant, we can use the equation for potential energy stored in a spring:
Potential Energy = (1/2) * k * x^2
Where:
k = spring constant (in N/m)
x = displacement from equilibrium position (in meters)
In this case, the mass is attached to the spring and when released, it reaches a maximum speed of 5 m/s. This means that all the potential energy stored in the spring has been converted into kinetic energy. We can equate the potential energy to the kinetic energy to find the value of x:
Potential Energy = Kinetic Energy
(1/2) * k * x^2 = (1/2) * m * v^2
Given:
mass (m) = 200 g = 0.2 kg
displacement (x) = 15 cm = 0.15 m
speed (v) = 5 m/s
Substituting the values into the equation, we get:
(1/2) * k * (0.15)^2 = (1/2) * 0.2 * 5^2
Simplifying:
k * (0.15)^2 = 0.2 * 5^2
k * 0.0225 = 0.2 * 25
k * 0.0225 = 5
k = 5 / 0.0225
k ≈ 222.22 N/m
Therefore, the spring constant is approximately 222.22 N/m.
To find the spring constant (k), we can use the equation for potential energy stored in a spring. The potential energy stored in a spring is given by the formula:
Potential Energy = 0.5 * k * (compression)^2
In this problem, the compression (x) is given as 15 cm, which is equivalent to 0.15 m. The mass (m) is given as 200 g, which is equivalent to 0.2 kg. The speed (v) of the mass is given as 5 m/s.
Using the equation for potential energy and setting it equal to the kinetic energy of the mass, we can solve for the spring constant (k).
Potential Energy = 0.5 * k * (compression)^2
Kinetic Energy = 0.5 * m * (velocity)^2
0.5 * k * (0.15)^2 = 0.5 * 0.2 * (5)^2
To solve for k, we can rearrange the equation and substitute the given values:
k = (0.5 * 0.2 * (5)^2) / (0.15)^2
k = (0.2 * 25) / 0.0225
k = 4 / 0.0225
k ≈ 177.78 N/m
Therefore, the spring constant is approximately 177.78 N/m.