Potential Energy

ENG1081 Physics for Engineering

Potential Energy, U

Consider the following scenarios:• Riding on a roller coaster• Jumping on a trampoline

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Work and Potential Energy

• We find that at instances where U is max, K is at minimum (and vice versa)

• Since K = W• U = -W

When U positive, K negativeWhen U negative, K positive

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Conservative and Non-Conservative Forces

This would be a good time to introduce two separate groups of Forces

Conservative Forces– Energy Transfer between Forces are Reversible

Non-Conservative Forces– Non-Reversible Transfer of Energy between Forces

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Testing for A Conservative Force

If it Passes Both of these Conditions:

1. In a Closed Path, the Net Work done on a Round Trip is 0.

2. Between Points A and B, the Work done is Independent of the Path taken

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Expression for U

• We consider a generalised Work equation

• And because U = -W

𝑊=∫𝑥 𝑖

𝑥𝑓

𝐹 (𝑥 )𝑑𝑥

∆𝑈=−∫𝑥 𝑖

𝑥 𝑓

𝐹 (𝑥 )𝑑𝑥

Force that Varies with x

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Gravitational Potential

∆𝑈=−∫𝑥 𝑖

𝑥 𝑓

𝐹 (𝑥 )𝑑𝑥 • Consider only vertical displacement (y)

• Take upwards direction as +ve

• Fg = -mg𝑈=𝑚𝑔𝑦

The higher the Object is, the more Gravitational Potential Energy it Contains 7

Elastic Potential

∆𝑈=−∫𝑥 𝑖

𝑥 𝑓

𝐹 (𝑥 )𝑑𝑥 The Spring Force, Fs = -kx

𝑈=12 𝑘𝑥

2

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Conservation of Energy

∆𝐾=𝑊∆𝑈=−𝑊⇒∆𝐾=−∆𝑈⇒𝐾 2−𝐾 1=−(𝑈 2−𝑈 1)

𝐾 2+𝑈 2=𝐾 1+𝑈 1

∆𝐸𝑚𝑒𝑐=∆𝐾 +∆𝑈=0

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Force and Potential Energy

• We can find U from F(x)

• How to we determine F(x) from U?

∆𝑈=−∫𝑥 𝑖

𝑥 𝑓

𝐹 (𝑥 )𝑑𝑥

∆𝑈=−𝑊=−𝐹 (𝑥 )∆ 𝑥

𝐹 (𝑥 )=− 𝑑(𝑈)𝑑𝑥 Check:

Gravity

Spring

𝐹 𝑔=−𝑑 (𝑚𝑔𝑥 )𝑑𝑥

𝐹 𝑠=−𝑑 (12 𝑘𝑥

2)

𝑑𝑥

Negative of the slope of plot U versus x 10

Spring Curves

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Note: In reading of these graphs, it is convenient to ignore the negative value of the Spring Force and treat it as any regular Force

Potential Energy Curve

x = 0

x +ve

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Potential Energy Curve

Turning Point

Turning Point

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Equilibrium Points

When the force acting on the object is zero, the object is said to be in a state of Equilibrium• Stable Equilibrium – located at minimums, will

return if the object is displaced slightly• Unstable Equilibrium – located at maximums,

will move away if the object is displaced slightly• Neutral Equilibrium – located at plateaus,

where the net force equals zero.

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Equilibrium Points

• Stable Equilibrium at x3 and x5.

• Unstable Equilibrium at x4.

• Neutral Equilibrium at x1 and x6.

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Conservation of Energy with Friction Involved

Consider a scenario where you are pushing a cart up a slope.

Work, W = K + U = Emec With Friction, W = Emec + Ethermal

In both cases, Conservation of Energy appliesWork, W = E = Emec + Ethermal

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