PHYS 218 Chapter 7
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Potential Energy
Possibility of work to be done (represented by ). 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.
Gravitational Energy
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_g = mgy = -\Delta U}
Elastic Energy
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_e = \frac{1}{2} k(x-l_0)^2}
Net Forces
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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?
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{k\Delta x^2}{2} + \frac{m{v_i}^2}{2} + mgy_i+W_{non-conservative} = mgh}
Forces and Potential Energy
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} F_x &= -\frac{dU}{dx} \\ F_y &= -\frac{dU}{dy} \\ F_z &= -\frac{dU}{dz} \\ F &= - \nabla U \end{align} }
Example
Potential Energy = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U=mgy}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_e} J of energy, then the total area of the graph is bounded by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=y_e}
Distance from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} -axis to graph is potential energy, distance from graph to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_e}
HARD EXAMPLE
An object is at height Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} on an inclined plane at angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} and an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} component of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} with a friction coefficient of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_k} . Once it gets to the bottom, it moves across a level, frictionless plane, then goes over a circular bump of radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} and flies off the track.
- Find the acceleration Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} on the inclined plane.
- Find the time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_1} at which the object hits the frictionless plane.
- Find the angle Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} at which the object flies off
1. Acceleration of inclined plane
Draw a free body diagram.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum F = \begin{cases}\hat{x} & mg \sin \varphi = ma_x \\ \hat{y} & N-mg \cos{\varphi} = ma_y = 0\end{cases}}
Using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F_y} , we can find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N=mg \cos{\varphi}} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_x = g(\sin{\varphi}-\mu_k \cos{\varphi})}
2. Velocity at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_1}
Use kinematics or conservation of energy
Kinematics
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} v^2 & = {v_0}^2 + 2a \Delta x \\ v & = \sqrt{2g(\sin{\varphi} - \mu_k\cos{\varphi})\sqrt{h^2+d^2}} \end{align}}
Conservation of Energy
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} K_i+U_i+\ldots+W_nc & = K_f+U_f \\ mgh - \mu_k mg \cos{\varphi} \sqrt{h^2+d^2} & = \frac{1}{2} m v^2 \\ 2g(h-\mu_kg\cos{\varphi}\sqrt{h^2+d^2} & = v^2 & \mbox{same as other method} \end{align}}
3. Angle of liftoff
Draw a free-body diagram (use Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} for coordinate system)
The object will leave when the normal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N=0}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum F = \begin{cases} \hat{r} & N-mg \cos{\theta} = m\frac{{v_\theta}^2}{R} \\ \hat{\varphi} & mg \sin{\theta} = m a_\theta \end{cases}}
Using conservation of energy to find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_\theta}
,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_k} makes an angle of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} with the ground and ends at a height Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_f} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} K_i+U_i+W_{non-conservative} &= K_f+U_f \\ 0+0+\mu_kN\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{align}}
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_k = 0.4} with the box. What is the velocity of the bucket after it descends 2 m?
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} K_i+U_i+W_{nc}=K_f+U_f \\ -\mu_kN(2)=\frac{1}{2}m_2v^2+\frac{1}{2}m_1v^2-m_2g(2) \end{align}}
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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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).
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -N-mg=-m\frac{v^2}{R}}
If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N=0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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