Atoms in a solid are not stationary, but vibrate about their equilibrium positions. Typically, the frequency of vibration is about f = 4.55 x 1012 Hz, and the amplitude is about 2.69 x 10-11 m. For a typical atom, what is its (a) maximum speed and (b) maximum acceleration?
To find the maximum speed and maximum acceleration of an atom vibrating in a solid, we can use the concepts of simple harmonic motion.
In simple harmonic motion, the displacement of an object is given by the equation:
x(t) = A * cos(2πft)
where:
x(t) is the displacement of the object at time t,
A is the amplitude of the motion,
f is the frequency of the motion,
and cos is the cosine function.
To find the maximum speed and maximum acceleration, we need to differentiate the displacement equation with respect to time.
(a) Maximum Speed:
The velocity of the object is given by the derivative of the displacement:
v(t) = dx(t)/dt = -2πfA * sin(2πft)
The maximum speed occurs when the absolute value of the velocity is at its highest. In this case, it will be when the sine function reaches its maximum value of 1. Therefore, the maximum speed of the atom is:
v_max = 2πfA
Substituting the given values:
v_max = 2π * (4.55 x 10^12 Hz) * (2.69 x 10^-11 m)
v_max ≈ 7.70 x 10^1 m/s
So, the maximum speed of the atom is approximately 77.0 m/s.
(b) Maximum Acceleration:
The acceleration of the object is given by the derivative of the velocity:
a(t) = dv(t)/dt = -4π^2f^2A * cos(2πft)
The maximum acceleration occurs when the absolute value of the acceleration is at its highest. In this case, it will be when the cosine function reaches its maximum value of 1. Therefore, the maximum acceleration of the atom is:
a_max = 4π^2f^2A
Substituting the given values:
a_max = 4π^2 * (4.55 x 10^12 Hz)^2 * (2.69 x 10^-11 m)
a_max ≈ 1.82 x 10^23 m/s^2
So, the maximum acceleration of the atom is approximately 1.82 x 10^23 m/s^2.