Solve Rigid Body Rotation Problem: Find x dot

[TheDude710]

Hi all. I am in the middle of a ridig body rotation problem. Anyways, I have x double dot = sin(x) and i need to find what x dot is. x is a function of time 't' and I just have no idea how to get it. so it is d^2x(t)/dt^2=sin(x(t)) where x(t) is a function and i need to find dx(t)/dt.

Any help would be great, thanks!

 

[learningphysics]

Let a=dx/dt

da/dt=d^2x/dt^2

Therefore
da/dt = sin(x)

(da/dx)(dx/dt) = sin(x)

(da/dx)(a) = sin(x)

a da = sin(x) dx

And you can get a by integrating both sides...

 

[thepatientmental]

To solve this problem, we can use the chain rule of differentiation. Let's start by rewriting the given equation as:

d^2x/dt^2 = sin(x(t))

We can then apply the chain rule to find dx/dt:

d/dt(d^2x/dt^2) = d/dt(sin(x(t)))

By the chain rule, the left side becomes:

d^2x/dt^2 * dx/dt = cos(x(t)) * dx/dt

We can rearrange this to solve for dx/dt:

dx/dt = (d^2x/dt^2) / cos(x(t))

Substituting the given equation for d^2x/dt^2, we get:

dx/dt = sin(x(t)) / cos(x(t))

Finally, we can simplify this by using the trigonometric identity for tangent:

dx/dt = tan(x(t))

Therefore, x dot (dx/dt) is equal to tan(x(t)). I hope this helps you solve your rigid body rotation problem!