PHYS 218 Chapter 7

« previous | Wednesday, March 7, 2011 | next »

Potential Energy

Possibility of work to be done (represented by $U$). Measured in Joules (J)

Gravitational potential energy: greater height → greater potential energy

Elastic potential energy: stretching/compressing a spring more → greater potential energy

Work-Energy Theorem

(See PHYS 218 Chapter 6#Work-Energy Theorem→)

Work of all forces is equivalent to the difference in Kinetic Energy.

$\begin{array}{rl}W& =K_2-K_1\\ \end{array}$

Gravitational Energy

$U_g=mgy=-\mathrm{\Delta }U$

Elastic Energy

$U_e=\frac{1}{2}k(x-l_0{)}^2$

Net Forces

$W_1+W_2+W_3=K_2-K_1$

Any W's that can be replaced by ΔU's are considered "conservative forces". All others are considred "non-conservative forces".

Result:

$\underset{\text{mechanical energy}}{\underbrace{K_i+U_{1i}+U_{2i}+\dots }}+W_{nc}=K_f+U_{1f}+U_{2f}+\dots$

Example

An object is stored in front of a compressed spring. The object will be pushed by the spring and move up a ramp. What is the max height?

$\frac{k\mathrm{\Delta }{x}^2}{2}+\frac{m{v_i}^2}{2}+mg{y}_i+W_{non-conservative}=mgh$

Forces and Potential Energy

$\begin{array}{rl}{F}_x& =-\frac{dU}{dx}\\ F_y& =-\frac{dU}{dy}\\ F_z& =-\frac{dU}{dz}\\ F& =-\mathrm{\nabla }U\end{array}$

Example

Potential Energy = $U=mgy$

$F=\mathrm{\nabla }U=(-\frac{dU}{dx},-\frac{dU}{dy},-\frac{dU}{dz})=(0,-mg,0)$

Energy Diagrams

Graph the energy equations (results in parabola)

If a system is given to have $y_e$ J of energy, then the total area of the graph is bounded by $y=y_e$

Distance from $x$-axis to graph is potential energy, distance from graph to $y_e$

HARD EXAMPLE

An object is at height $h$ on an inclined plane at angle $\phi$ and an $x$ component of $d$ with a friction coefficient of ${\mu }_k$. Once it gets to the bottom, it moves across a level, frictionless plane, then goes over a circular bump of radius $R$ and flies off the track.

1. Find the acceleration $a$ on the inclined plane.
2. Find the time $t_1$ at which the object hits the frictionless plane.
3. Find the angle $\theta$ at which the object flies off

1. Acceleration of inclined plane

Draw a free body diagram.

$\sum F=\{\begin{array}{ll}\hat{x}& mg\sin \phi =m{a}_x\\ \hat{y}& N-mg\cos \phi =m{a}_y=0\end{array}$

Using $F_y$, we can find that $N=mg\cos \phi$.

$a_x=g(\sin \phi -{\mu }_k\cos \phi )$

2. Velocity at $t_1$

Use kinematics or conservation of energy

Kinematics

$\begin{array}{rl}{v}^2& ={v_0}^2+2a\mathrm{\Delta }x\\ v& =\sqrt{2g(\sin \phi -{\mu }_k\cos \phi )\sqrt{h^2+d^2}}\end{array}$

Conservation of Energy

$\begin{array}{rl}{K}_i+U_i+\dots +W_{n}c& =K_f+U_f\\ mgh-{\mu }_{k}mg\cos \phi \sqrt{h^2+d^2}& =\frac{1}{2}m{v}^2\\ 2g(h-{\mu }_{k}g\cos \phi \sqrt{h^2+d^2}& =v^2& \text{same as other method}\end{array}$

3. Angle of liftoff

Draw a free-body diagram (use $r$ and $\phi$ for coordinate system)

The object will leave when the normal $N=0$

$\sum F=\{\begin{array}{ll}\hat{r}& N-mg\cos \theta =m\frac{{v_{\theta }}^2}{R}\\ \hat{\phi }& mg\sin \theta =m{a}_{\theta }\end{array}$

Using conservation of energy to find $v_{\theta }$,

$\frac{1}{2}m{v_i}^2=\frac{1}{2}m{v_f}^2+mg(R+\cos \theta )$

---

Monday, March 7, 2011

Another Problem

A ramp with friction coefficient of ${\mu }_k$ makes an angle of $\phi$ with the ground and ends at a height $h$. A rocket at the base of the ramp will fire its engine until it leaves the ramp.

What is the force of the engine if the velocity at height $L$ is $v_f$.

Note: Friction and the force of the engine are non-conservative forces

$\begin{array}{rl}{K}_i+U_i+W_{non-conservative}& =K_f+U_f\\ 0+0+{\mu }_{k}N\frac{H}{\sin \phi }+F_{engine}d& =\frac{1}{2}m{v_f}^2+mgL\\ {\mu }_{k}mg\cos \phi \frac{H}{\sin \phi }+F_e\frac{H}{\sin \phi }& =\frac{1}{2}m{v_f}^2+mgL\end{array}$

Yet Another Problem

An 80 kg box sits on a ledge with a rope attached to a pulley and ultimately to a 65 kg bucket. The ground has a friction coefficient ${\mu }_k=0.4$ with the box. What is the velocity of the bucket after it descends 2 m?

$\begin{array}{r}{K}_i+U_i+W_{nc}=K_f+U_f\\ -{\mu }_{k}N(2)=\frac{1}{2}{m}_2{v}^2+\frac{1}{2}{m}_1{v}^2-m_{2}g(2)\end{array}$

Loop-the-loop

"How high should the ball be to make a full circle without falling off the rail?"

"make a full circle"

Circular motion: $a=\frac{v^2}{R}$

"without falling off the rail"

Normal has to be greater than zero (the point at which an object breaks free of circular motion is when the normal is zero).

$-N-mg=-m\frac{v^2}{R}$

If $N=0$, $g=\frac{v^2}{R}$

MOST IMPORTANT!

When is the mechanical energy of a system conserved?

As long as the work of the non-conservative forces is equal to 0