444g mass vibrates according to the equation x = 0.348 sin (5.82 t) where x is in meters and t is in seconds. Determine the total energy. Determine the kinetic energy when x is 14.0cm. Determine the potential energy when x is 14.0cm.
total energy = 0.912J
PE = 1.53 x 10^3 J
KE = 0.910 J
To determine the total energy of the vibrating mass, we need to find both the kinetic and potential energy components.
First, let's determine the kinetic energy at x = 14.0 cm. Kinetic energy (KE) is given by the equation:
KE = (1/2) * m * v^2
where m is the mass of the object and v is the velocity.
To find the velocity, we need to differentiate the position equation with respect to time (t):
v = dx/dt = d(0.348 sin(5.82t))/dt
v = 0.348 * d(sin(5.82t))/dt
v = 0.348 * 5.82 cos(5.82t)
Now, at x = 14.0 cm = 0.14 m, we can substitute the value of x into the position equation to find the time (t):
0.14 = 0.348 sin(5.82t)
Rearranging the equation, we have:
sin(5.82t) = 0.14 / 0.348
t = arcsin(0.4023) / 5.82
Now that we have the time (t), we can substitute it into the velocity equation to find the velocity (v):
v = 0.348 * 5.82 cos(5.82t)
v = 0.348 * 5.82 cos(5.82 * arcsin(0.4023) / 5.82)
Once we have the velocity, we can calculate the kinetic energy (KE):
KE = (1/2) * m * v^2
KE = (1/2) * 444g * [0.348 * 5.82 cos(5.82 * arcsin(0.4023) / 5.82)]^2
Now, let's determine the potential energy at x = 14.0 cm. Potential energy (PE) is given by the equation:
PE = (1/2) * k * x^2
where k is the spring constant. We know that x = 14.0 cm = 0.14 m, and the equation gives us the spring constant.
Finally, to determine the total energy (TE), we add the kinetic and potential energy components together:
TE = KE + PE
Substituting in the calculated values, we can find the total energy of the vibrating mass.