Physics - Chapter 7

Lesson 28: Radian Measurement

Notes

LEARNING TARGET JOURNAL

  • What is a radian?
  • What is the radius of the unit circle?
  • What is the circumference of the unit circle?
  • How many degrees are in a circle?
  • So _____ radians is equivalent to _____ degrees.

NOTES

  • Translational motion occurs between one point and another.
  • Rotational motion is when a rigid object maintains its shape while rotating around a fixed axis.
  • Circular motion is a type of rotational motion that occurs for any object following part of, or an entire circular path. AQUrYb1.gif

http://upload.wikimedia.org/wikipedia/commons/thumb/3/3d/Radian_cropped_color.svg/220px-Radian_cropped_color.svg.png

{\theta ^o} \cdot \left( {\frac{\pi }{{{{180}^o}}}} \right) = \theta {\rm{ rad}}

\theta {\rm{ rad}} \cdot \left( {\frac{{{{180}^o}}}{\pi }} \right) = {\theta ^o}

  • The units for measuring angles in radians. (rad)
  • Although, many times the measures are left unit-less as radian is often an understood angle measure.

  • Recall that from the unit circle.
  • To convert revolutions to radians, multiply the revolutions by .https://mathtestpreparation.com/Lessons/TrignoRadianFig3.gif
  • Angular motion is a vector quantity that account for the direction rotation.
  1. clockwise – which we will account for as negative in direction
  2. counterclockwise – which we will account for as positive in direction

EXAMPLE 1 – Perform the following conversions.

  1. 125° to radians
  1. 17 radians to degrees
  1. 6800 RPM to rad/s

HOMEWORK -


Physics - Chapter 7

Lesson 29: Angular Kinematics

Notes

LEARNING TARGET JOURNAL

  • What is the difference between angular speed and linear speed?
  • What are some similarities of angular motion and linear motion?

  • List the any factors you can think of that help angular speed generate faster linear speeds?
  • How can it be said that something is said to be traveling at 120 mph when it is not moving in front of us?


NOTES

  • Angular displacement () is the change in position as an object rotates.

\Delta \theta  = \frac{{\Delta s}}{r}

  • s is the arc length
  • r is the length of the radius of the circle
  • The SI unit for angular displacement is radians. (rad)

http://t1.gstatic.com/images?q=tbn:ANd9GcSv_N-Azz3pbkr81hgT9HEDbvAElBwchPwcj7YkUigFvlODRi-2

  • Angular velocity () it is the rate of change in angular displacement.                {\omega _{avg}} = \frac{{{\theta _f} - {\theta _i}}}{{{t_f} - {t_i}}}
  • The SI base unit for angular velocity is  .
  • Another common unit for angular velocity is revolutions per minute. (RPM)

{\alpha _{avg}} = \frac{{{\omega _f} - {\omega _i}}}{{{t_f} - {t_i}}}

  • Angular acceleration () is the rate of change in angular velocity.        
  • The SI base unit for angular acceleration is  .

  • All these quantities fit the same relationships as their translational counterparts from Chapters 2 & 3.

EXAMPLE 1 – Find the indicated value.

  1. While riding on a carousel that is rotating clockwise, a child travels through an arc length of 11.5 m. If the child’s angular displacement is -165°, what is the radius of the carousel?
  1. A car tire rotates with an average angular speed of 29 rad/s. In what time interval will the tire rotate 3.5 times?
  1. A figure skater begins spinning counterclockwise at an angular speed of 4.0π rad/s. During a 3.0 s interval, she slowly pulls her arms inward and finally spins at 8.0π rad/s. What is her average angular acceleration during this time interval?

EXAMPLE 2 – Use the angular kinematic equations to solve the following.

  1. Find the angular acceleration of a spinning amusement-park ride that initially travels at 0.50 rad/s then accelerates to 0.60 rad/s during a 0.50 s time interval.
  1. The wheel on an upside-down bicycle moves through 11.0 rad in 2.0 s. What is the wheel’s angular acceleration if its initial angular speed is 2.0 rad/s?
  1. A remote-controlled car’s wheel accelerates at 22.4 rad/s2 begins with an angular speed of 10.8 rad/s, what is the wheel’s angular speed after exactly three full turns?

HOMEWORK -


Physics - Chapter 7

Lesson 30: Tangential Velocity and Acceleration

Notes

LEARNING TARGET JOURNAL

  • What happens to the rotational speed at the end of a lever when it is shortened?
  • TRUE or FALSE: Every point on a wheel travels with the same angular speed?         TRUE         FALSE
  • TRUE or FALSE: Every point on a wheel travels with the same tangential speed?         TRUE         FALSE

NOTES

  • Tangential speed is the equivalent instantaneous linear speed of an object along its circular path.http://hendrix2.uoregon.edu/~dlivelyb/phys101/images/L7-3.gif
  • All points on the circle at any radius travel with the same angular speed.

{v_t} = r\omega

  • The variable for tangential speed is .http://t1.gstatic.com/images?q=tbn:ANd9GcSpc1pnJI8q2N-x2ifSKC84xhWt-zeMh5O0-TvRYIs34XQDTHXe
  • The SI base unit for tangential speed is the same as linear velocity, .

  • Tangential acceleration is the equivalent instantaneous linear acceleration an object experiences during circular motion. https://figures.boundless.com/13714/medium/figure-11-01-02a.jpeg
  • The direction of the acceleration is acting perpendicular to the radius of rotation.
  • The direction of the acceleration will match the tangential speed when speeding up, and be opposite the tangential speed when slowing down.
  • Again, objects farther away from the axis of rotation will encounter a larger tangential acceleration due to the larger magnitude of velocity being on a larger circumference .http://www.astro.ucla.edu/~malkan/astro8/physics1a/rotvel.gif
  • The variable for tangential acceleration is .                                
  • The SI base unit for tangential acceleration is the same as translational acceleration, .

EXAMPLE 1 – Solve the following.

  1. Find the tangential speed of a ball swung at a constant angular speed of 5.0 rad/s on a rope that is 5.0 m long.
  1. A woman passes through a revolving door with a tangential speed of 1.8 m/s. If she is 0.80 m from the center of the door, what is the door’s angular speed?

EXAMPLE 2 – Solve the following.

  1. Find the tangential acceleration of a person standing 9.5 m from the center of a spinning amusement-park ride that has an angular acceleration of 0.15 rad/s2.
  1. A dog on a merry-go-round undergoes a 1.5 m/s2 linear acceleration. If the merry-go-round’s angular acceleration is 1.0 rad/s2 dog, how far from the axis of rotation is the dog?

EXAMPLE 3 – A point on the rim of a 0.40 m radius rotating disc has a tangential speed of 4.0 m/s. What is the tangential speed of a point 0.30 m from the center of the same disc?

  1. First we need to know the angular speed of the disc.
  1. Since every spot on that disc has the same angular speed, use that value to find the tangential speed at 0.30 m.

HOMEWORK -


Physics - Chapter 7

Lesson 31: Centripetal Acceleration

Notes

LEARNING TARGET JOURNAL

What is the difference between centripetal motion and centrifugal motion?



NOTES

  • Centrifugal characterizes the motion of a rotating particle that is “fleeing” the center of rotation.
  • Centripetal characterizes the motion of a rotating particle that is “seeking” the center of rotation.

  • Centripetal acceleration is the rate of change of the tangential velocity.
  • The formula for centripetal acceleration can be used in two forms.
  • Using tangential speed                {a_c} = \frac{{{v_t}^2}}{r}
    LEAVE SPACE FOR PROOF



                    
  • Using angular speed                {a_c} = r{\omega ^2}
    LEAVE SPACE FOR PROOF


    https://figures.boundless.com/13715/large/figure-11-01-03a.jpe
  • The variable for centripetal acceleration is .
  • The SI unit for centripetal acceleration is .

  • The total acceleration of the object is the sum of the tangential acceleration and centripetal acceleration acting at any moment along the circular path.
  • It is not a linear sum but a two dimensional sum, because centripetal acceleration acts along the radius of rotation and tangential acceleration acts perpendicular to that radius.

EXAMPLE 1 – Solve the following.

  1. A 1.2 kg stone is attached to a 1.3 m line and swung in a circle. If the stone has a linear speed of 13 m/s, what is the centripetal acceleration?
  1. A piece of clay sits 0.20 m from the center of a potter’s wheel. If the potter spins the wheel at an angular speed of 20.5 rad/s, what is the magnitude of the centripetal acceleration of the piece of clay on the wheel?

EXAMPLE 2 – Solve the following.

  1. A young boy swings a yo-yo horizontally above his head so that the yo-yo has a centripetal acceleration of 250 m/s2. If the yo-yo’s string is 0.50 m long, what is the yo-yo’s tangential speed?
  1. A race car moves along a circular track at an angular speed of 0.512 rad/s. If the car’s centripetal acceleration is 15.4 m/s2 between the car and the center of the track, what is the distance between the car and the center of the track?

HOMEWORK -


Physics - Chapter 7

Lesson 32: Centripetal Force

Notes

LEARNING TARGET JOURNAL

  1. FILL IN THE BLANK: There is no such thing as _________________ force.
  2. When two candles are lit on either end of a rotating rod as in Dr. Julius Sumner Miller’s Candle Paradox,
  1. What do you expect to happen to the candle flames?
  2. What actually happened?
  3. But why does the governor’s bob spin out?

NOTES

  • Centrifugal force is a fictitious force because it violates Newton’s 3rd Law of Motion by not having a reaction force.
  • The feeling of being pulled away from the center is caused by inertial effects of traveling in along an ever changing path.
  • Newton’s 2nd Law leads us to the following formulas:
  • using tangential velocity                {F_c} = m\frac{{{v_t}^2}}{r}        
  • using angular velocity                        {F_c} = mr{\omega ^2}
  • The variable for centripetal force we use is , with the c indicating centripetal force.http://www.hometrainingtools.com/media/images/art/CentripetalForce.jpg
  • The SI unit for centripetal force is the same as any other force, newtons. (N)

  • Centripetal force is some net force that is acting in a centripetal manner. For instance, Image result for centripetal force gravity
  • the tension in a string attached to a massive object rotating in a circle.
  • the force of gravity acting on a satellite as it orbits the Earth.


EXAMPLE 1 – Solve the following.

  1. A 90.0 kg person rides a spinning amusement park ride that has an angular speed of 1.15 rad/s. If the radius of the ride is 11.5 m, what is the magnitude of the force that maintains the circular motion of the person.
  1. A pilot is flying a small plane at 30.0 m/s in a circular path with a radius of 100.0 m. If a force of 635 N is needed to maintain the pilot’s circular motion, what is the pilot’s mass?

HOMEWORK -


Physics - Chapter 7

Lesson 33: Centripetal Force From Friction

Notes

LEARNING TARGET JOURNAL

  1. What are the three types of circular acceleration?
  2. Describe how each affects the motion of an object traveling in uniform circular motion.


NOTES

  • Imagine a coin resting on a rotating disc. What holds the coin from sliding off the disc as it rotates? http://t2.gstatic.com/images?q=tbn:ANd9GcRTNXjyuK1xHFKiSJKjuQB2PlQM9xtlWZwurQJ4QerUVhANt8CMiA
  • Upon inspection of a free-body diagram, the static friction between the coin and the disc is acting centripetal force.http://philschatz.com/physics-book/resources/Figure_07_03_02a.jpg

{F_c} = \mu  \cdot {F_n} = m\frac{{{v_t}^2}}{r}

  • The same can be for a car rounding a flat curve.

  • For banked curves, the normal force helps to aid the centripetal force.

EXAMPLE 1 – Solve.

  1. A 13,500 N car traveling at 50.0 km/h rounds a curve of radius 2.00 × 102 m. Find the minimum coefficient of static friction between the tires and the road that will allow the car to round the curve safely.
  1. A 2.00 × 103 kg car rounds a circular turn of radius 20.0 m. If the road is flat and the coefficient of static friction between the tires and the road is 0.70, how fast can the car go without skidding?

HOMEWORK -



Physics - Chapter 7

Lesson 34: Newton’s Universal Law of Gravitation

Notes

LEARNING TARGET JOURNAL

Why does the International Space Station rotate?



NOTES

  1. directly proportional to the product of their masses, and
  2. inversely proportional to the square of the separation between themhttp://titan.bloomfield.edu/facstaff/dnicolai/images/ImagesPhy-105/Chapter%204/lesson4.jpg

{F_g} = {\bf{G}}\frac{{{m_1} \cdot {m_2}}}{{{r^2}}}

  • This separation, r, is measured from their centers of mass
  • The m here stands for mass of each object
  • And G is called the constant of universal gravitation

  • The force on each object is the same magnitude, but always attractive so as to follow Newton’s 3rd Law of Motion.

EXAMPLE 1 – Solve the following.

  1. A 90.0 kg person stands 1.00 m from a 60.0 kg person sitting on a bench nearby. What is the magnitude of the gravitational force between them?
  1. Mars has a mass of about 6.4 × 1023 kg, and its moon Phobos has a mass of about 9.6 × 1015 kg. If the magnitude of the gravitational force between the two bodies is 4.6 × 1015 N, how far apart are Mars and Phobos?

EXAMPLE 2 – Without using actual mass and separation values, determine how a change in value(s) will affect the gravitational force between two objects.

  1. If both masses are tripled, what happens to the force?
  1. If the masses are not changed, but the distance of separation is tripled, what happens to the force?
  1. If one of the masses is doubled, the other remains unchanged, and the distance of separation is quadrupled, show what happens to the force.

HOMEWORK -


Physics - Chapter 7

Lesson 35: Orbital Velocity

Notes

LEARNING TARGET JOURNAL

  1. What is the orbital velocity of a low level satellite?
  2. What is the orbital velocity of the high level satellites?
  3. Why do they differ?

NOTES

  • Newton’s Universal Law of Gravitation now explains that the attracting force of gravity is the centripetal force

{\bf{G}}\frac{{{m_1} \cdot {m_2}}}{{{r^2}}} = {m_2}\frac{{{v_t}^2}}{r}

  • The most common use for this equation is to find the orbiting speed of a satellite  at any given height  above the Earth.

Leave space for proof.

  • The formula to find the orbital velocity is:                                        {v_{orb}} = \sqrt {\frac{{{\bf{G}} \cdot {M_E}}}{r}}
  • The remaining mass is the mass of the Earth.
  • Keep in mind, the orbiting radius  is measured from the center of the Earth.
  • The radius of the Earth is .


EXAMPLE 1 – Solve the following.

  1. A satellite has a mass of 100 kg and is in a circular orbit of 2.00 x 106 m above the surface of the Earth. Determine the instantaneous velocity of the satellite in its orbit.
  1. A satellite moves in a circular orbit around the Earth at a speed of 5000 m/s. Determine the satellite’s altitude above the surface of the Earth.

HOMEWORK -