PHYS 218 Chapter 7

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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.

${\begin{aligned}W&=K_{2}-K_{1}\\\end{aligned}}$

Gravitational Energy

$U_{g}=mgy=-\Delta U$

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:

$\underbrace {K_{i}+U_{1i}+U_{2i}+\ldots } _{\mbox{mechanical energy}}+W_{nc}=K_{f}+U_{1f}+U_{2f}+\ldots$

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?

${\frac {k\Delta x^{2}}{2}}+{\frac {m{v_{i}}^{2}}{2}}+mgy_{i}+W_{non-conservative}=mgh$

Forces and Potential Energy

${\begin{aligned}F_{x}&=-{\frac {dU}{dx}}\\F_{y}&=-{\frac {dU}{dy}}\\F_{z}&=-{\frac {dU}{dz}}\\F&=-\nabla U\end{aligned}}$

Example

Potential Energy = $U=mgy$

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

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 $\varphi$ 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.

$\sum F={\begin{cases}{\hat {x}}&mg\sin \varphi =ma_{x}\\{\hat {y}}&N-mg\cos {\varphi }=ma_{y}=0\end{cases}}$

Using $F_{y}$, we can find that $N=mg\cos {\varphi }$.

$a_{x}=g(\sin {\varphi }-\mu _{k}\cos {\varphi })$

2. Velocity at $t_{1}$

Use kinematics or conservation of energy

Kinematics

${\begin{aligned}v^{2}&={v_{0}}^{2}+2a\Delta x\\v&={\sqrt {2g(\sin {\varphi }-\mu _{k}\cos {\varphi }){\sqrt {h^{2}+d^{2}}}}}\end{aligned}}$

Conservation of Energy

${\begin{aligned}K_{i}+U_{i}+\ldots +W_{n}c&=K_{f}+U_{f}\\mgh-\mu _{k}mg\cos {\varphi }{\sqrt {h^{2}+d^{2}}}&={\frac {1}{2}}mv^{2}\\2g(h-\mu _{k}g\cos {\varphi }{\sqrt {h^{2}+d^{2}}}&=v^{2}&{\mbox{same as other method}}\end{aligned}}$

3. Angle of liftoff

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

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

$\sum F={\begin{cases}{\hat {r}}&N-mg\cos {\theta }=m{\frac {{v_{\theta }}^{2}}{R}}\\{\hat {\varphi }}&mg\sin {\theta }=ma_{\theta }\end{cases}}$

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

${\frac {1}{2}}m{v_{i}}^{2}={\frac {1}{2}}m{v_{f}}^{2}+mg(R+\cos {\theta })$

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Monday, March 7, 2011

Another Problem

A ramp with friction coefficient of $\mu _{k}$ makes an angle of $\varphi$ 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

${\begin{aligned}K_{i}+U_{i}+W_{non-conservative}&=K_{f}+U_{f}\\0+0+\mu _{k}N{\frac {H}{\sin {\varphi }}}+F_{engine}d&={\frac {1}{2}}m{v_{f}}^{2}+mgL\\\mu _{k}mg\cos {\varphi }{\frac {H}{\sin {\varphi }}}+F_{e}{\frac {H}{\sin {\varphi }}}&={\frac {1}{2}}m{v_{f}}^{2}+mgL\end{aligned}}$

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?

${\begin{aligned}K_{i}+U_{i}+W_{nc}=K_{f}+U_{f}\\-\mu _{k}N(2)={\frac {1}{2}}m_{2}v^{2}+{\frac {1}{2}}m_{1}v^{2}-m_{2}g(2)\end{aligned}}$

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