A 1.55 kg mass attached to a spring oscillates with a period of 0.375 s and an amplitude of 16.5 cm.
(a) Find the total mechanical energy of the system.
(b) Find the maximum speed of the mass.
x=A•sinωt
v =dx/dt = A•ω•cosωt
v(max) = A•ω,
where ω =2•π/T.
E (total) = m•(ω• A)^2/2 .
To find the total mechanical energy of the system, we need to consider the potential energy and the kinetic energy. The total mechanical energy (E) is the sum of these two energies.
(a) First, let's find the potential energy of the system at the maximum displacement. The potential energy (PE) of a spring is given by the equation:
PE = 0.5 * k * x^2
where k is the spring constant and x is the displacement from the equilibrium position.
In this case, the amplitude (A) is given as 16.5 cm, which is equivalent to 0.165 m. Since the maximum displacement occurs at the amplitude, we can substitute this value into the equation:
PE = 0.5 * k * (0.165)^2
Next, let's find the kinetic energy of the system at the maximum speed. The kinetic energy (KE) of an object in simple harmonic motion is given by the equation:
KE = 0.5 * m * v^2
where m is the mass of the object and v is the velocity.
The velocity can be calculated using the formula for the period (T) of oscillation:
T = 2π * sqrt(m / k)
Rearranging this equation, we can solve for v:
v = 2π * (A / T)
Substituting the given values:
v = 2π * (0.165 / 0.375)
Now, we can substitute the mass and velocity into the equation for kinetic energy:
KE = 0.5 * (1.55) * (v)^2
To find the total mechanical energy, we add the potential and kinetic energies:
E = PE + KE
Plug in the values to find your answer.
(b) The maximum speed of the mass is given by the formula we derived earlier:
v = 2π * (A / T)
Substitute the given values of the amplitude (A) and the period (T), and calculate the result to find the maximum speed.