the motion of a body is described in simple harmonic motion as x=Cos(wt).when the body is 0.2m from the mid of its path , velocity is 3m/s and when it is 0.8m from the center of its path, its velocity is 1m/s ,determine:
1.the period
2.greatest acceleration
To determine the period and greatest acceleration of the body in simple harmonic motion, we can use the given equation x = Cos(wt), where x represents the displacement of the body, w represents the angular frequency, and t represents time.
1. Period of motion:
The period of simple harmonic motion is the time taken for the body to complete one full cycle. In this case, one full cycle is from the initial position, through maximum displacement to the other extreme position, and back to the initial position.
Let's start with the given information: when the body is 0.2m from the mid-point of its path, its velocity is 3m/s. We can determine the angular frequency (w) using this information.
At x = 0.2m, we have:
x = Cos(wt)
0.2 = Cos(wt)
To find the angular frequency (w), we can take the inverse cosine (or cos^-1) of both sides:
cos^-1(0.2) = wt
w = cos^-1(0.2) / t
Alternatively, you can use the known relationship between velocity and displacement in simple harmonic motion:
v = -w * Sin(wt)
At x = 0.2m, v = 3m/s:
3 = -w * Sin(wt)
Dividing both sides by -w:
-3/w = Sin(wt)
Since the Sin function is dimensionless (-1 ≤ Sin(wt) ≤ 1), -3/w must also be dimensionless. Therefore, the denominator w has units of inverse time (1/t). This confirms that w represents the angular frequency.
Similarly, when the body is 0.8m from the center of its path, its velocity is 1m/s:
1 = -w * Sin(wt)
Now we have two equations:
-3/w = Sin(wt) (equation 1)
1 = -w * Sin(wt) (equation 2)
Dividing equation 2 by equation 1, we get:
(1) / (-3/w) = -w * Sin(wt) / Sin(wt)
-3/w^2 = -w
Simplifying the equation further:
w^2 = 3
From this equation, we can solve for w:
w = √(3)
The period (T) can be found using the formula:
T = 2π / w
Substituting the value of w, we have:
T = 2π / √(3)
2. Greatest acceleration:
The acceleration in simple harmonic motion can be determined using the equation:
a = -w^2 * x
Substituting the given displacement (x = 0.8m) and the angular frequency (w = √(3)), we have:
a = -3 * 0.8
a = -2.4 m/s^2
Therefore, the greatest acceleration is 2.4 m/s^2 (since acceleration in SHM is always negative).
To summarize:
1. The period of the motion is T = 2π / √(3)
2. The greatest acceleration of the body is 2.4 m/s^2.