A mass of 0•5 kg is attached to a spring. The mass is then displaced from its equilibrium position by 5cm and released. It's speed as it passes the equilibrium position is 50cm/s.
A)calculate the spring constant of the spring.
B)calculate the maximum acceleration of the block.
C)calculate the speed of the block when x=3cm.
C)calculate the pariod of oscillation.
D)calculate the acceleration when x=2cm.
E)calculate the total mechanical energy of the system.
To solve this problem, we will use the principles of simple harmonic motion and the equations derived from Hooke's Law. Hooke's Law states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.
A) To calculate the spring constant (k), we can use the equation for potential energy in a spring:
Elastic Potential Energy (PE) = 1/2 * k * x^2
In this case, the mass (m) is displaced by 5 cm from the equilibrium position, so x = 0.05 m, and the speed is given as 50 cm/s.
The potential energy at the maximum displacement equals the kinetic energy when the mass passes the equilibrium position:
Potential Energy (PE) = Kinetic Energy (KE)
1/2 * k * x^2 = 1/2 * m * v^2
Substituting the given values:
1/2 * k * (0.05)^2 = 1/2 * 0.5 * (0.5)^2
0.0125k = 0.03125
k = 0.03125 / 0.0125
k = 2.5 N/m
Therefore, the spring constant is 2.5 N/m.
B) To calculate the maximum acceleration of the block, we use the equation for maximum acceleration in simple harmonic motion:
Maximum acceleration (amax) = ω^2 * A
Where ω is the angular frequency and A is the amplitude.
We can find ω using the equation:
Angular frequency (ω) = 2π / T
And T is the period of oscillation.
C) To calculate the speed of the block when x = 3 cm, we can use the equation for velocity in simple harmonic motion:
Velocity (v) = ω * √(A^2 - x^2)
D) To calculate the period of oscillation, we can use the equation for the period of a mass-spring system:
Period of oscillation (T) = 2π * √(m / k)
E) To calculate the acceleration when x = 2 cm, we can use the equation for acceleration in simple harmonic motion:
Acceleration (a) = -ω^2 * x
Finally, to calculate the total mechanical energy of the system, we can use the equation:
Total Mechanical Energy (E) = Potential Energy (PE) + Kinetic Energy (KE)
Now, let's calculate the values step by step.