From equation x= A sin wt, prove that the acceleration of SHM is a= -w^2x
w is omega actually..
2.using relationship between velocity, v and displacement, x prove that v=w(A^2-x^2)^1/2
Why no answer?
To prove that the acceleration in Simple Harmonic Motion (SHM) is given by a= -w^2x, we will differentiate the equation x= A sin wt twice with respect to time (t).
Given: x= A sin wt
First, let's find the first derivative of x with respect to time (t) to get velocity (v).
Differentiating x with respect to t, we get:
v = d/dt (A sin wt)
Using the chain rule, the derivative of sin(wt) with respect to t is cos(wt) * w.
So, v = A * cos(wt) * w
Next, let's find the second derivative of x with respect to time to get acceleration (a).
Differentiating v (A * cos(wt) * w) with respect to t, we get:
a = d/dt (A * cos(wt) * w)
Using the chain rule again, the derivative of cos(wt) with respect to t is -sin(wt) * w.
So, a = -A * sin(wt) * w^2
Since sin(wt) = x/A, we can substitute it in the acceleration equation.
a = -w^2 * (x/A) * A
Simplifying the equation, we get:
a = -w^2 * x
Therefore, we have proven that the acceleration of an object undergoing Simple Harmonic Motion (SHM) is given by a= -w^2x.
Moving on to the second part of the question:
To prove the relationship between velocity (v) and displacement (x) as v = w(A^2 - x^2)^1/2, we can start with the given equation of SHM, x= A sin wt.
Let's differentiate x with respect to time (t) to get velocity (v).
Differentiating x with respect to t, we get:
v = d/dt (A sin wt)
Using the chain rule, the derivative of sin(wt) with respect to t is cos(wt) * w.
So, v = A * cos(wt) * w
Now, we need to express cos(wt) in terms of displacement (x). For that, we can use the Pythagorean identity: cos^2(wt) = 1 - sin^2(wt).
From the equation x= A sin wt, we can rewrite it as sin^2(wt) = (x/A)^2.
Now, substituting this into the equation cos^2(wt) = 1 - sin^2(wt), we get:
cos^2(wt) = 1 - (x/A)^2
Taking the square root on both sides, we have:
cos(wt) = (1 - (x/A)^2)^1/2
Substituting this into the equation for velocity (v), we get:
v = A * (1 - (x/A)^2)^1/2 * w
Simplifying further, we have:
v = w * (A^2 - x^2)^1/2
Therefore, we have proven that the relationship between velocity (v) and displacement (x) in Simple Harmonic Motion (SHM) is v=w(A^2-x^2)^1/2.