A 2.20 kg mass attached to a spring oscillates with a period of 0.440 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:
(a) The total mechanical energy of the system consists of two parts: the potential energy stored in the spring and the kinetic energy of the mass.
The potential energy of a spring is given by the formula:
PE = (1/2)kx^2,
where PE represents the potential energy, k is the spring constant, and x is the displacement (amplitude). In this case, the amplitude is given as 16.5 cm, which can be converted to meters by dividing by 100:
x = 16.5 cm = 0.165 m.
The spring constant (k) can be determined using the formula for the period of oscillation (T) of a mass-spring system:
T = 2π√(m/k),
where T is the period, m is the mass of the object, and k is the spring constant. Rearranging this equation gives us:
k = (4π^2m) / T^2.
Now we can substitute the values into the formula:
k = (4π^2 * 2.20 kg) / (0.440 s)^2.
Solving this equation will give you the value of k.
Next, we can calculate the potential energy using the formula:
PE = (1/2)kx^2.
Substitute the known values of k and x to find the potential energy.
To find the kinetic energy, we use the equation:
KE = (1/2)mv^2,
where KE represents the kinetic energy and v is the velocity. At maximum displacement, when the spring's potential energy is at its maximum, the entire potential energy is converted into kinetic energy. Therefore, the potential energy at maximum displacement is equal to the kinetic energy.
So, the total mechanical energy (E) can be expressed as:
E = PE + KE,
where E denotes the total mechanical energy.
Substituting the potential and kinetic energy values into the equation will give you the total mechanical energy of the system.
(b) To find the maximum speed of the mass:
At maximum displacement, the energy is fully converted from potential energy into kinetic energy. Therefore, the maximum speed occurs when the mass passes through the equilibrium position.
We can apply the principle of conservation of mechanical energy to find the maximum speed (vmax) of the mass:
E = KE + PE,
where E represents the total mechanical energy of the system, KE is the kinetic energy, and PE is the potential energy.
At the maximum displacement, the potential energy is at its maximum value, which is equal to the total mechanical energy. Therefore:
E = PE + KE = PE + PE = 2 * PE.
Knowing the potential energy, we can write the equation for the total mechanical energy as:
E = 2 * PE.
Now, we need to find the equation for the potential energy at the maximum displacement:
PE = (1/2)kx^2.
Substituting the known values of k and x into the equation will give you the potential energy at maximum displacement.
Finally, substitute this value of potential energy into the equation E = 2 * PE to find the total mechanical energy.
To find the maximum speed (vmax), we can use the equation for kinetic energy:
KE = (1/2)mv^2.
Substituting the total mechanical energy (found in part (a)) and the mass of the object into the equation will allow you to solve for vmax by rearranging and solving for v.