Gravitational Potential Energy © Simon Porter 2007 © Simon Porter 2007 Gravitational Potential Energy

How much GPE?

How much GPE? GPE = mgh?

How much GPE? GPE = mgh?

How much GPE? GPE = mgh?

We do know that the GPE must be decreasing. But where is the GPE zero? How much GPE? GPE = mgh? We do know that the GPE must be decreasing. But where is the GPE zero?

How much GPE? GPE = mgh? We do a little physicists trick. We take the GPE at infinity to be zero! That means that it has negative GPE at distance closer than infinity!

Gravitational potential energy © Simon Porter 2007 © Simon Porter 2007 Gravitational potential energy Gravitational potential energy at a point is defined as the work done to move a mass from infinity to that point.

Gravitational potential energy © Simon Porter 2007 © Simon Porter 2007 Gravitational potential energy Gravitational potential energy at a point is defined as the work done to move a mass from infinity to that point. I’ve come from infinity! m M R

Gravitational potential energy © Simon Porter 2007 © Simon Porter 2007 Gravitational potential energy Gravitational potential energy at a point is defined as the work done to move a mass from infinity to that point. Work done = force x distance The force however is changing as the mass gets closer I’ve come from infinity! m M R

Gravitational potential energy © Simon Porter 2007 © Simon Porter 2007 Gravitational potential energy R R [ ] R - GMm r - GMm R W = Fdr = GMmdr r2 = = I’ve come from infinity! m M R

Gravitational potential energy © Simon Porter 2007 © Simon Porter 2007 Gravitational potential energy Gravitational potential energy at a point is defined as the work done to move a mass from infinity to that point. Ep = -GMm r Ep is always negative

Gravitational Potential © Simon Porter 2007 © Simon Porter 2007 Gravitational Potential It follows that the Gravitational potential at a point is the work done per unit mass on a small point mass moving from infinity to that point. It is given by V = -GM r Ep = mV Note the difference between gravitational potential energy (J) and Gravitational potential (J.kg-1)

Moving masses in potentials If a mass is moved from a position with potential V1 to a position with potential V2, work = m(V2 – V1) = mΔV (independent of path) V2 V1

Equipotential surfaces/lines

Equipotential surfaces/lines

Field and equipotentials Equipotentials are always perpendicular to field lines. Diagrams of equipotential lines give us information about the gravitational field in much the same way as contour maps give us information about geographical heights.

Field strength = potential gradient In fact it can be shown from calculus that the gravitational field is given by the potential gradient (the closer the equipotential lines are together, the stronger the field) g = -dV dr

Let’s stop and read! Pages 127 to 130 Pages142 to 151

Escape speed Imagine throwing a ball into the air © Simon Porter 2007

Escape speed It falls to the ground (Doh!) © Simon Porter 2007

Escape speed What happens if you throw harder? © Simon Porter 2007

Escape speed It goes higher and takes longer to return. © Simon Porter 2007 © Simon Porter 2007 Escape speed It goes higher and takes longer to return.

Escape speed It goes higher and takes longer to return. Ouch! © Simon Porter 2007 © Simon Porter 2007 Escape speed It goes higher and takes longer to return. Ouch!

© Simon Porter 2007 © Simon Porter 2007 Escape speed The kinetic energy of the ball changes to gravitational potential energy as the ball rises. This in turn turns back into kinetic energy as the ball falls again.

© Simon Porter 2007 © Simon Porter 2007 Escape speed How fast would you have to throw the ball so that it doesn’t come back? (i.e. goes to “infinity” or escapes the gravitational field of the earth)

© Simon Porter 2007 © Simon Porter 2007 Escape speed At “infinity”, it gravitational energy is given by Ep = -GMm/r = zero when r is infinite

© Simon Porter 2007 © Simon Porter 2007 Escape speed Energy conservation tells us that it must therefore have zero energy to start with if it is to escape the earth’s gravity. i.e. KE + GPE = 0

Escape speed i.e. KE + GPE = 0 ½mv2 + -GMem/Re = 0 © Simon Porter 2007 © Simon Porter 2007 Escape speed i.e. KE + GPE = 0 ½mv2 + -GMem/Re = 0 (where Re is the radius of the earth) ½mv2 = GMem/Re v = √2GMe/Re

In reality the escape velocity of the earth is bigger than this. WHY? © Simon Porter 2007 © Simon Porter 2007 Escape speed v = √2GM/Re v = √(2 x 6.67 x 10-11 x 5.98 x 1024)/6.38 x 106 v = 12000 m.s-1 I can’t throw that fast! In reality the escape velocity of the earth is bigger than this. WHY?

Let’s try some questions! Page 153 Q7, 13

Hold on! Isn’t electricity similiar? © Simon Porter 2007

Gravitational Potential © Simon Porter 2007 © Simon Porter 2007 Gravitational Potential The Gravitational potential at a point is the work done per unit mass on a small point mass moving from infinity to that point. It is given by V = -GM r Ep = mV Note the difference between gravitational potential energy (J) and Gravitational potential (J.kg-1)

Electrical Potential V = W q Uel = qV © Simon Porter 2007 © Simon Porter 2007 Electrical Potential The Electrical potential at a point is the work done per unit charge on a small positive test charge moving from infinity to that point. It is given by V = W q Uel = qV Scalar quantity Note the difference between electrical potential energy (J) and Electrical potential (J.C-1)

Moving charges in potentials If a charge is moved from a position with potential V1 to a position with potential V2, work = q(V2 – V1) = qΔV (independent of path) V2 V1

Gravitational potential energy © Simon Porter 2007 © Simon Porter 2007 Gravitational potential energy Gravitational potential energy at a point is defined as the work done to move a mass from infinity to that point. Ep = -GMm r Ep is always negative

Electrical potential energy © Simon Porter 2007 © Simon Porter 2007 Electrical potential energy Electrical potential energy at a point is defined as the work done to move a positive charge from infinity to that point. Uel = kQq r

Equipotential surfaces/lines Ep = -GMm r

Equipotential surfaces/lines

Field and equipotentials Equipotentials are always perpendicular to field lines. Diagrams of equipotential lines give us information about the gravitational field in much the same way as contour maps give us information about geographical heights.

Field strength = potential gradient In fact it can be shown from calculus that the gravitational field is given by the potential gradient (the closer the equipotential lines are together, the stronger the field) E = dV dr

Gravitation Electricity © Simon Porter 2007 © Simon Porter 2007 From “Physics for the IB Diploma” K.A.Tsokos (Cambridge University Press) Gravitation Electricity Acts on Mass (always +?) Charge (+ or -) Force F = GM1M2/r2 Attractive only, infinite range F = kQ1Q2/r2 Attractive or repulsive, infinite range Relative strength 1 1042 Field g = GM/r2 E = kQ/r2 Potential V = -GM/r V = kQ/r Potential energy Ep = -GMm/r Ep = kQq/r

Let’s try some questions Pages 307 Questions 2, 4, 5, 6, 11, 12.