Mock Final

Problem 1:

Part a: Calculate $F(x)$

We know that

$$F=MA=M\frac{dv}{dt}$$

So we need to take the time derivative of the velocity, to find acceleration, to find force. We have velocity as a function of x, which can be expressed as a function of time.

$$V(x(t))=\frac{\alpha }{x}$$

We’re gonna do some “funny math” to take the time derivative of velocity

$$\frac{dv}{dt}=\frac{dv}{dt}\frac{dx}{dx}$$

Multiply by “1” ($\frac{dx}{dx}$)

$$\frac{dv}{dt}=\frac{dv}{dx}\frac{dx}{dt}$$

We know the time derivative of position is velocity

$$\frac{dv}{dt}=\frac{dv}{dx}v$$

Now we just have to take the position derivative of velocity

$$\frac{dv}{dt}=-\frac{\alpha }{x^2}v$$

Subbing in $\frac{dv}{dt}$ for a, and $v$ for $v$

$$a=-\frac{\alpha }{x^2}\frac{\alpha }{x}$$

Plug into newtons second law

$$F(x)=m\left[-\frac{{\alpha }^2}{x^3}\right]$$

Part B: Calculate X(t)

We can solve this by treating it like a separable differential equation

$$\frac{dx}{dt}=\frac{\alpha }{x}$$

Separate

$$xdx=\alpha dt$$

Integrate

$$\int xdx=\int \alpha dt$$ $$\frac{x^2}{2}=\alpha t$$

Solve for x

$$x(t)=\sqrt{2\alpha t}$$

The rest of this problem is pretty trivial.

Problem 2:

For part a: Finding the Lagrange Equations of Motion

We need to define the degrees of freedom for this problem, there exists 2. $\theta$: angular position of mass $m$ on the table $r$: radial position of mass m on table, and coupled to mass M below table

Now, we need to find our equations for kinetic and potential energy

$$T=\frac{1}{2}m(r\dot{\theta }{)}^2+\frac{1}{2}m{\dot{r}}^2+\frac{1}{2}M{\dot{r}}^2$$ $$U=Mg(l-r)$$ $$L=T-U$$ $$L=\frac{1}{2}m(r\dot{\theta }{)}^2+\frac{1}{2}m{\dot{r}}^2+\frac{1}{2}M{\dot{r}}^2-Mg(l-r)$$

Use E-L equations Solving for $\theta$

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{\theta }}-\frac{\partial L}{\partial \theta }=0$$ $$\frac{d}{dt}\left(2\frac{1}{2}m{r}^2\dot{\theta }\right)-0=0$$

Product Rule

$$2m\dot{r}\dot{\theta }+m{r}^2\ddot{\theta }=0$$

Simplify

$$2\dot{r}\dot{\theta }+r\ddot{\theta }$$

Solving for $r$

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{r}}-\frac{\partial L}{\partial r}=0$$ $$\frac{d}{dt}\left(\frac{1}{2}2m\dot{r}+\frac{1}{2}2M\dot{r}\right)-\left[\frac{1}{2}2mr{\dot{\theta }}^2+Mgr\right]=0$$ $$(m+M)\ddot{r}-mr{\dot{\theta }}^2+Mg=0$$

Part B: Solving for equilibrium separation

When $\dot{r}$ and $\ddot{r}$ is 0, the radius is constant

Setting $\ddot{r}$ to 0:

$$-mr{\dot{\theta }}^2+Mg=0$$ $$Mg=mr{\dot{\theta }}^2$$ $$\frac{Mg}{m{\dot{\theta }}^2}=r_0$$

Problem 3:

Problem 4:

(this will be in Cartesian coordinates tho so don’t worry too much)

Problem 5:

Problem 6:

Will likely be a plucked, struck string… I couldn’t find a good example, so just go through HW 11