A body vibrates in simple harmonic motion with an amplitude of 5 cm and a frequency of 0.50 Hz.
a) Calculate:
i) the maximum acceleration
ii) the maximum velocity
iii) the magnitude of the acceleration and velocity when the body is displaced 10 mm from its equilibrium position.
b) State the values of the constants A (in m) and W (in rad*s^-1) in the equation y = AsinWt which describes the motion of the body.
x = .05 sin [2 pi (.5)t]
= .05 sin 3.14159 t
v = .05 pi cos pi t
a = -.05 pi^2 sin pi t
max a = .05 pi^2
max v = .05 pi
10 mm = 1 cm = .01 m
x = .01 = .05 sin 3.14 t
sin 3.14 t = .20
3.14 t = 11.5 degrees = .201 radians
t = .0641 seconds
v = .05 pi cos 11.5 degrees
a = -.05 pi^2 sin 11.5 degrees
a)
i) To calculate the maximum acceleration, we use the formula:
amplitude (A) = maximum acceleration (amax) / angular frequency (ω)
Since the amplitude is given as 5 cm and the angular frequency is given as 0.50 Hz, we need to convert the amplitude to meters and rearrange the formula to find amax:
amplitude = 5 cm = 0.05 m (1 cm = 0.01 m)
angular frequency = 0.50 Hz
amax = amplitude × angular frequency
amax = 0.05 m × 0.50 s^-1
amax = 0.025 m s^-2
Therefore, the maximum acceleration is 0.025 m s^-2.
ii) To calculate the maximum velocity, we use the formula:
amplitude (A) = maximum velocity (vmax) / angular frequency (ω)
Again, we rearrange the formula to find vmax:
amplitude = 5 cm = 0.05 m
angular frequency = 0.50 Hz
vmax = amplitude × angular frequency
vmax = 0.05 m × 0.50 s^-1
vmax = 0.025 m s^-1
Therefore, the maximum velocity is 0.025 m s^-1.
iii) To calculate the magnitude of acceleration and velocity when the body is displaced 10 mm from its equilibrium position, we need to use the formula for displacement:
y = Asin(ωt)
where A is the amplitude, ω is the angular frequency, and t is the time.
Since the body is displaced 10 mm = 0.01 m, we can substitute the values into the equation:
0.01 = A × sin(ωt)
To find the magnitude of acceleration, we differentiate the equation with respect to time:
a = A × ω^2 × cos(ωt)
Similarly, to find the magnitude of velocity, we differentiate the equation with respect to time:
v = A × ω × cos(ωt)
Substituting the given values, A = 0.05 m and ω = 2π × f = 2π × 0.50 Hz = 1 π rad/s, and solving for a and v when t = 0:
a = A × ω^2 × cos(ωt)
a = 0.05 × (π)^2 × cos (0)
a = 0.05 × 9.87 × 1
a = 0.4935 m/s^2
v = A × ω × cos(ωt)
v = 0.05 × π × cos (0)
v = 0.05 × π × 1
v = 0.157 m/s
Therefore, when the body is displaced 10 mm from its equilibrium position, the magnitude of acceleration is 0.4935 m/s^2, and the magnitude of velocity is 0.157 m/s.
b)
In the equation y = Asin(Wt), the constant A represents the amplitude of the motion in meters. In this case, A = 0.05 m.
The constant W represents the angular frequency of the motion in radians per second. The angular frequency can be calculated using the formula:
angular frequency (W) = 2π × frequency (f)
Given that the frequency is 0.50 Hz, we can substitute it into the formula:
W = 2π × 0.50 Hz
W = π rad/s
Therefore, the values of the constants A and W in the equation y = Asin(Wt) are A = 0.05 m and W = π rad/s.