Physics 221 SI Exam 2 Review
15. Potential Energy and Potential Energy curves
a. Equations for these types of problems
Gravitionional Potential Energy = U = 𝑚𝑔ℎ
i. Described by a change in height and does no depend on the path that the object
takes, only whether it has a positive or negative change in height
1 2
𝑘𝑥
2
ii. Typically described by a changed in elongation or compression of a spring
In General for a force in the x direction : 𝑈 = − ∫ 𝐹 𝑑𝑥 and therefore
Elastic(Spring)Potential Energy = U =
𝑑𝑈
− 𝑑𝑥 = 𝐹 so in a graph of potential energy the Force can be found by the
derivative of the curve
16. Non-conservative Work
a. Work done by an external force that removes energy from the system most often this is
Friction
Described as A loss or change in total energy through a process
∆𝐸 = ∆𝐾𝐸 + ∆𝑈 = 𝑊𝑛𝑜𝑛−𝑐𝑜𝑛𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑣𝑒 or 𝑊𝑛𝑜𝑛−𝑐𝑜𝑛𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑣𝑒 = 𝐹𝐹𝑟𝑖𝑐𝑡𝑖𝑜𝑛 ∆𝑥 Force
times the distance is it applied over
If only conservative forces involved ∆𝐸 = ∆𝐾𝐸 + ∆𝑈 = 0 aka energy is conserved
17. Energy Diagrams
a. As displayed in the problem from earlier find if the particle is in equilibrium by
𝑑𝑈
determining if Force is 0 by - 𝑑𝑥 = 𝐹 = 0
𝑑2 𝑈
b. Then is it stable or unstable equilibrium (concave up or concave down) - 𝑑𝑥 2 = +𝑜𝑟 − if
positive (concave up) it is stable, if negative (concave down) its unstable
c. When looking at an Energy diagram it can sometimes be analyzed as if it were a ball
rolling up and down hills. Where the initial position helps to determine total mechanical
energy of the ball and it can only travel through the hills if it has enough energy to
overcome the peak of a hill.
18. Linear Momentum and collisions
a. Linear Momentum 𝑝 = 𝑚𝑣 momentum conserved if 𝑝𝑖 = 𝑝𝑓
b. A change in momentum is described as Impulse created by a net external force
𝐽 = ∆𝑝 = ∫ 𝐹𝑒𝑥𝑡 𝑑𝑡
c. Collisions
i. Perfectly Elastic collision- Kinetic energy is conserved
ii. Inelastic Collision-Kinetic Energy decreases (when objects deform in collision)
iii. “Perfectly Inelastic Collisions- Kinetic Energy decreases but can be identified by
combining Kinetic NRG and momentum equations (Objects stick together after
collision)
iv. Explosions- Kinetic energy increases, Bodies break into parts, explosion
mechanism provides more Kinetic energy
v. Super-elastic collision- KE increases, some internal energy is transformed into
KE due to collision
d. For elastic collisions another useful equation to allow to solve problems is the relative
velocity equation
𝑣𝑖1 − 𝑣𝑖2 = −(𝑣𝑓1 − 𝑣𝑓2 )
20. Center of Mass
∑ 𝑚𝑟
a. the equation for the center of mass of a system is 𝑟𝑐𝑚 = ∑𝑖 𝑚𝑖𝑟𝑖
𝑖
𝑖 𝑖
b. which can help us coincidentally find the acceleration and velocity of center of mass by
∑ 𝑚𝑣
∑ 𝑚𝑎
taking the derivative with respect to time obtaining 𝑣𝑐𝑚 = ∑𝑖 𝑚𝑖𝑣𝑖and 𝑎𝑐𝑚 = ∑𝑖 𝑚𝑖𝑎𝑖
𝑖
𝑖 𝑖
𝑖
𝑖 𝑖
c. Then multiplying the velocity equation by the sum of the masses can help us describe
how the momentum of the center of mass can be found
𝑝𝑡𝑜𝑡𝑎𝑙 = ∑ 𝑝𝑖 = ∑ 𝑚𝑖 𝑣𝑖 = 𝑚𝑡𝑜𝑡𝑎𝑙 𝑣𝑐𝑚
𝑖
Center of mass example run through….
𝑖
21. Moment of Inertia & Rigid Body Motion
a. Moment of inertia can be described as ______________ or ____________ when
a body is rotation about the same axis, a certain distance parallel to the axis of
rotation given some standard moments about the center of mass of objects you
will need to use those often
b. Another addition to conservation of energy is rotational kinetic energy
___________ now objects moving down an incline with rotation must now store
energy in rotation and linear kinetic energy in addition to gravitational potential
energy
c. Torque often an important concept to look at when analyzing problems is 𝑇 = 𝑟 ×
𝐹 or 𝑇𝑛𝑒𝑡 = 𝐼𝛼 in order to start rotation a net torque must be present
d. A net torque can provide work by 𝑊 = ∫ 𝑇𝑑𝜃 or a Torque multiplied by a Δθ arc
it is applied through
e.
24. Angular Momentum
a. Equations related to
i. 𝐿 = 𝑟 × 𝑝
ii. 𝐿 = 𝐼𝜔
iii. 𝑇 =
𝑑𝐿
𝑑𝑡
iv. When net torque is zero angular momentum is conserved and 𝐿𝑖 = 𝐿𝑓
25. Gravitation
a. Equations
26. Fluids
a. Equations
Book problems