A 0.025 kg object vibrates in SHM at the end of a spring. If the maximum displacement of the object is 0.030 cm, and its period is 0.50s, what is the maximum acceleration of the object?
Max acceleration is w^2 times maximum displacement, where w is the angular frequency, which is 2*pi*f.
its 56 cxm
To find the maximum acceleration of the object in simple harmonic motion (SHM), we can use the formula:
a_max = (2π/T)^2 * x_max
Where:
a_max = maximum acceleration
T = period
x_max = maximum displacement
Given:
T = 0.50 s
x_max = 0.030 cm = 0.030 * 0.01 m = 0.003 m
Substituting the values into the formula:
a_max = (2π/0.50)^2 * 0.003
Calculating:
a_max = (2 * 3.1416 / 0.50)^2 * 0.003
a_max ≈ 4π^2 * 0.003
a_max ≈ 4 * 3.1416^2 * 0.003
a_max ≈ 4 * 9.8696 * 0.003
a_max ≈ 0.11806 m/s^2
Therefore, the maximum acceleration of the object is approximately 0.11806 m/s^2.
To find the maximum acceleration of the object vibrating in Simple Harmonic Motion (SHM), we can use the formula:
a = -ω^2 * x
Where:
a = maximum acceleration
ω = angular frequency
x = maximum displacement
First, let's calculate the angular frequency (ω) using the formula:
ω = 2π / T
Where:
T = period
Given that the period (T) is 0.50s, we can substitute it into the formula:
ω = 2π / 0.50s
Simplifying this equation, we get:
ω = 4π rad/s
Now we can substitute the values of ω and x into the formula for maximum acceleration:
a = -ω^2 * x
a = - (4π rad/s)^2 * 0.030 cm
Note that we need to convert the displacement from centimeters to meters, as the SI unit for acceleration is meters per second squared (m/s^2).
To convert cm to m, we divide by 100 since there are 100 centimeters in a meter:
a = - (4π rad/s)^2 * (0.030 cm / 100)
Simplifying further, we get:
a ≈ -9.48 m/s^2
So the maximum acceleration of the object is approximately -9.48 m/s^2. Note that the negative sign indicates that the acceleration is in the opposite direction of the displacement.