PHY 131 Midterm 2 Study Sheet PDF

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Summary

Summarizes the material for the 2nd midterm of the class...

Description

Energy and Momentum -

Energy is the quantitative property that must be transferred to a body or physical system to perform work on the body Work is the energy transferred to or from an object via the application of force along with a displacement The law of conservation of energy states that the total energy of an isolated system remains the same Equations: - Kinetic Energy:

-

-

2

𝑚𝑣

1 2

2

-

𝑃𝐸 =

-

𝑘 = 𝑆𝑝𝑟𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑎𝑛𝑑 𝑥 = 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑘𝑥

Gravitational Potential Energy:

𝑃𝐸 = 𝑚𝑔ℎ

Conservation of Energy with no nonconservative forces:

-

1 2

Spring Potential Energy:

-

𝐾𝐸 =

𝐾𝐸𝑖 + 𝑆𝑃𝐸𝑖 + 𝐺𝑃𝐸𝑖 =𝐾𝐸𝑓 + 𝑆𝑃𝐸𝑓 + 𝐺𝑃𝐸𝑓

Conservation of Energy with nonconservative forces:

-

𝑊𝑛𝑐 = (𝐾𝐸𝑓 + 𝑆𝑃𝐸𝑓 + 𝐺𝑃𝐸𝑓) − (𝐾𝐸 + 𝑆𝑃𝐸 + 𝐺𝑃𝐸 ) 𝑖 𝑖 𝑖

-

-

Nonconservative Forces: - Friction - Applied Force - Push and Pull Action - Tension Force Momentum is the amount of motion of a moving object, measured as a product of its mass and velocity

-

-

𝑊𝑚𝑐 𝑖𝑠 𝑛𝑜𝑟𝑚𝑎𝑙𝑙𝑦 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 µ𝐹𝑛𝑥 / µ𝑚𝑔𝑥 (𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛)

𝑝 = 𝑚𝑣

Impulse is a certain amount of force you apply for a certain amount of time to cause a change in momentum nisis

- 𝐽 = ∆𝑝 = 𝐹∆𝑡 The Two Types of Collisions: -

Elastic Collisions: In a collision such as this one, the objects which collide remain separate after the collision In this situation, the total kinetic energy and momentum are conserved

-

Equations: - Conservation of momentum:

-

1 2

2

𝑚1𝑣1 +

𝑣1 − 𝑉1 =

2

𝑚2𝑣2 =

2'

1 2

1 2

𝑚 𝑣1 + 1

2'

𝑚 𝑣2 2

− (𝑣2 − 𝑉2)

Inelastic Collisions: - In a collision such as this one, the objects collide but then become one mass with one velocity and one mass - In this situation, only the momentum is conserved, not kinetic energy - Equations: - Conservation of momentum: -

-

1 2

Conservation of kinetic energy (simplified):

-

'

Conservation of kinetic energy: -

-

,

𝑚1𝑣1 + 𝑚2𝑣2 = 𝑚1𝑣1 + 𝑚 𝑣 2 2

𝑚1𝑣1 + 𝑚2𝑣2 = (𝑚1 + 𝑚2)𝑣

Elastic Collisions in the 2nd Dimension: - Behave similarly to elastic collisions, but instead the objects move in different directions - You have to use the conservation of momentum equation for both the x and y directions in this scenario - X:

-

,

'

,

'

-

𝑚1𝑣1𝑥 + 𝑚2𝑣2𝑥 = 𝑚1𝑣1𝑥 + 𝑚 𝑣 2 2𝑥

-

𝑚1𝑣1𝑦 + 𝑚2𝑣2𝑦 = 𝑚1𝑣1𝑦 + 𝑚 𝑣 2 2𝑦

Y:

Rotational Kinematics -

Angular Motion:

-

θ = Angular Displacement ω = Angular Velocity α = Angular Acceleration

-

Equations:

-

-

2

θ = θ0 + ω0𝑡 +

-

ω = ω0 + α𝑡

-

ω = ω0 + 2α(θ − θ0)

-

𝑑 = 𝑟θ 𝑣 = 𝑟ω 𝑎 = 𝑟α

2

α𝑡

2

Torque -

-

-

1 2

-

Torque is defined as similar to linear force, whereas it applies to rotational motion. Known as a force applied to an object at a point away from the point of rotation Equations:

-

τ = 𝐹𝑟 τ = 𝐼α

-

For these equations, the 𝑟 is the distance away from the point of rotation, in other words, if there is a force acting directly towards the point of rotation (the𝑟 equals 0), then it should not be included in the torque equation

Moment of Inertia -

Inertia is the resistance of any physical object to any change in its velocity Inertia is symbolized by the letter 𝐼 Useful Equations:

-

Continuous mass distribution:

-

Upper Limit: - 𝑙𝑜𝑛𝑔 * 𝐿 Lower Limit: - 𝑠ℎ𝑜𝑟𝑡 * 𝐿 R: - 𝑥 dm: -

𝑀 𝐿

𝑑𝑥

Density could replace M/L Density: 𝑝 = 𝑚 / 𝑉

Torque

-

Torque is the rotational equivalent to linear force A force applied to an object at a point away from the point of rotation will cause a torque A net torque applied to an object will cause an angular acceleration The greater the torque, the greater the tendency for the object to rotate Torque Equation:

-

τ = 𝐼α τ = 𝐹𝑟

Centripical Acceleration:

-

𝑎𝑐 = 𝑎/𝑅

Rotational Energy -

Basically the same concept as Energy and Momentum but with rotational energy thrown into the equation for the conservation of energy ¯\_(ツ)_/¯ Equations: - Rotational Energy:

- 𝐸

𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙

-

1 2

2

𝐼ω

Conservation of Energy with no nonconservative forces:

-

=

𝐾𝐸𝑖 + 𝑆𝑃𝐸𝑖 + 𝐺𝑃𝐸𝑖 +

1 2

2

𝐼ω𝑖 =𝐾𝐸 + 𝑆𝑃𝐸 + 𝐺𝑃𝐸 + 𝑓 𝑓 𝑓

2

𝐼ω𝑓

Conservation of Energy with nonconservative forces:

-

𝑊

𝑛𝑐

= (𝐾𝐸 + 𝑆𝑃𝐸 + 𝐺𝑃𝐸 𝑓

𝑓

𝑓

+

1 2

2

𝐼ω𝑓 ) − (𝐾𝐸 + 𝑆𝑃𝐸 + 𝐺𝑃𝐸 + 𝑖 𝑖 𝑖

Angular Momentum -

1 2

Basically the same concept as Momentum but with angular momentum Equations: - Angular Momentum / Point Objects:

1 2

𝐼ω

-

Conservation of Angular Momentum:

-

𝐿 = 𝑚𝑣𝑟 '

'

𝐼1ω1 + 𝐼2ω2 = 𝐼1ω1 + 𝐼2ω2

Conservation of Angular Momentum in an Inelastic Collision:

-

𝐼1ω1 + 𝐼2ω2 = (𝐼1 + 𝐼2)ω

Fluids -

Equations: Note: "𝐴" 𝑠𝑡𝑎𝑛𝑑𝑠 𝑓𝑜𝑟 𝑡ℎ𝑒 𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑎 𝑠ℎ𝑎𝑝𝑒 - Pressure: -

-

Bernoulli’s principle -

-

𝑃 = 𝐹/𝐴 𝑃 = 𝑚𝑔ℎ 𝑃1 +

-

2

𝑝𝑣2 + 𝑝𝑔ℎ2

𝑝1𝐴1𝑣1 =𝑝2𝐴2𝑣2 𝐴1𝑣1 =𝐴2𝑣2

Archimedes’ Principle

-

𝐹𝑏 = 𝑚𝑔 = 𝑝𝑔𝑉

- 𝑅𝑒𝑝𝑙𝑎𝑐𝑒 𝑚 𝑤𝑖𝑡ℎ 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 * 𝑣𝑜𝑙𝑢𝑚𝑒 Volume rate of flow - Gas and Liquids

-

𝑄 = 𝑝𝑣𝐴

Liquids Only

-

1 2

Liquids Only

-

-

2

𝑝𝑣1 + 𝑝𝑔ℎ1 = 𝑃2 +

- "𝑃" 𝑠𝑡𝑎𝑛𝑑𝑠 𝑓𝑜𝑟 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑛𝑑 "𝑝" 𝑠𝑡𝑎𝑛𝑑𝑠 𝑓𝑜𝑟 𝑓𝑙𝑢𝑖𝑑 𝑑𝑒𝑛𝑠𝑖𝑡𝑦 Continuity Equations - Gas and Liquids

-

-

1 2

𝑄 = 𝑣𝐴

Important Stuff:

-

𝑝𝑤𝑎𝑡𝑒𝑟 = 1000

-

𝑃𝑎𝑖𝑟 = 100, 000 𝑝𝑎𝑠𝑐𝑎𝑙𝑠 = 1𝑎𝑡𝑚

-

Gauge Pressure: - Guage pressure is the difference between the absolute pressure and the atmospheric pressure

-

𝑃𝑔 = 𝑃𝑇 − 𝑃𝑎𝑡𝑚...