A 1.55 kg mass attached to a spring oscillates with a period of 0.375 s and an amplitude of 16.5 cm.
Find the total mechanical energy of the system
Find the maximum speed of the mass
ω = 2•π/T
E = m• ω^2•A^2/2 = m• (2•π)^2•A^2/2• T^2
To find the total mechanical energy of the system, we need to know the equation for the mechanical energy of a mass-spring system.
The total mechanical energy (E) of a mass-spring system is the sum of the potential energy (PE) and the kinetic energy (KE):
E = PE + KE
The potential energy of a mass-spring system is given by the formula:
PE = (1/2) k x^2
where k is the spring constant and x is the displacement from the equilibrium position.
The kinetic energy of the mass is given by the formula:
KE = (1/2) m v^2
where m is the mass and v is the velocity of the mass.
First, let's find the potential energy. The spring constant (k) can be calculated using the formula:
k = (2π/T)^2 m
where T is the period of oscillation and m is the mass.
Substituting the given values, we have:
k = (2π/0.375)^2 * 1.55
Now, let's find the potential energy (PE) when the mass is at the maximum displacement (amplitude). The displacement (x) is given as 16.5 cm, which is 0.165 m.
PE = (1/2) k x^2
Substituting the values, we get:
PE = (1/2) * [(2π/0.375)^2 * 1.55] * (0.165)^2
Next, let's find the maximum speed of the mass. The maximum speed occurs when the displacement is zero (i.e., at the equilibrium position). At this point, all the mechanical energy is in the form of kinetic energy.
KE = E - PE
To find the maximum speed, we need to subtract the potential energy at the maximum displacement (PE) from the total mechanical energy (E).
Finally, once you have calculated the total mechanical energy and the maximum speed, you can provide the answers.