Topic 3.3: Gravitational Potential Energy

Name: _______________________________ Date: _______________ Period: _______

Before We Start: Yesterday's Recap

Last class: net work equals the change in kinetic energy, $W_{net}=\Delta K$. A book is lifted at constant speed from the floor to a shelf. Kinetic energy doesn't change, so the net work is zero. Then where did the energy you supplied to the book go?

Today's New Concepts

AP CED Alignment: Unit 3, Topic 3.3

Gravitational Potential Energy Belongs to a System

• Potential energy is stored by the arrangement of a system — gravitational potential energy belongs to the object–Earth system (or two masses), not to one object alone
• Near Earth's surface: $\Delta U_g=mg\Delta y$, where $\Delta y$ is the change in height of the object's center of mass
• Up ($\Delta y>0$) means $\Delta U_g>0$; down ($\Delta y<0$) means $\Delta U_g<0$

Worked Example: Two $2.0\text{ kg}$ spheres touch a ceiling $3.0\text{ m}$ above the floor and are lowered to the floor. Sphere 1 has radius $0.10\text{ m}$, Sphere 2 has radius $0.40\text{ m}$. Center of mass drops: $\Delta y=H-2r$, so $2.8\text{ m}$ and $2.2\text{ m}$. $|\Delta U_1|=(2.0)(10)(2.8)=56\text{ J}$; $|\Delta U_2|=(2.0)(10)(2.2)=44\text{ J}$. The smaller sphere loses more.

The Zero Line Is a Free Choice

• Choose where $U_g=0$ (floor, table, cliff bottom). That changes the value of $U_g$ but never $\Delta U_g$, since $\Delta y$ doesn't depend on where zero is placed
• $U_g$ follows height, not speed: an object falling at constant terminal speed has $\Delta K=0$ but still has $\Delta U_g<0$

Worked Example: A $2.0\text{ kg}$ object falls $10\text{ m}$ at constant speed. $\Delta U_g=(2.0)(10)(-10)=-200\text{ J}$, and $\Delta K=0$. The lost potential energy did not become kinetic energy (Topic 3.4 explains where it went).

The Universal Form: $U_g=-\dfrac{Gm_1m_2}{r}$

• $r$ is the center-to-center distance; $U=0$ at infinite separation, so $U$ is always negative
• Farther apart ($r\uparrow$) $\Rightarrow$ $U$ increases (rises toward zero). Closer ($r\downarrow$) $\Rightarrow$ $U$ decreases (more negative)
• Write $U$ for each situation and let constants cancel in ratios; watch for both mass and distance changing

Worked Example: Planet A: mass $m_0$, radius $r_0$. Planet B: mass $4m_0$, radius $2r_0$, same star. $U_A=-GMm_0/r_0$; $U_B=-GM(4m_0)/(2r_0)=-2GMm_0/r_0$. $U_A:U_B=1:2$.