physics 185f2013 lecture three - drexel universitytim/phys185/lecture_seven_nov_05_2013/w… ·...
TRANSCRIPT
Physics 185F2013 Lecture ThreeNov 5, 2013
Dr. Jones1
1Department of PhysicsDrexel University
November 5, 2013
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 1 / 24
Inertial Frames
Imagine you place your cellphone on the dashboard ofyour car. You take a sharpturn around a curve. Whathappens to the cell phone?Why?
In our previous work, weassumed that we were workingin a Non-inertial referenceframe.
= a frame which is notaccelerating
Earth is not a non-inertialreference frame.
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 2 / 24
Inertial Frames
Inertial Frame
A frame that is not accelerating.Newton’s laws will hold precisely insuch a frame. The Earth isaccelerating as it travels around thesun and as it rotates and wobbles,so it isn’t an inertial frame.
Fictitious Force
A fictitious force is a force which isseen in a non-inertial frame whichdoesn’t seem to hold to Newton’sLaws. Normal forces happenbetween two objects, fictious forceappears to act on an object alone–no second object can be identified.
Example
As you make a turn in a car,objects in the car will slide as ifacted upon by a force, though thereis no source.
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 3 / 24
Two observers, two different stories
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 4 / 24
Inertial Frames
Example
As you make a turn in a car,objects in the car will slide as ifacted upon by a force, though thereis no source.
Centripetal acceleration
a =v2
R(1)
Always along the radius connectinga body to its center of rotation.
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 5 / 24
Example 1
Engineers design on-ramps and off-ramps of major highways to have asignificant slant or “bank”. Why would they do this? Consider a rampwith speed-limit 30.0 mi/h (13.4 m/s) and a radius of curvature of35.0m. If you are designing this road, what angle would you bank thisturn with? Draw free body diagrams.
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 6 / 24
Example 2
Design a Ferris Wheel that can handle up to 1000 kg per car. Drawfree body diagrams.
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 7 / 24
Example 3
Suppose you are navigating a rocket in deep space and you want to goin a circle of radius 3,053 meters at a speed of 100 m/s. By how muchand in what direction should you accelerate the rocket?
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 8 / 24
Coriolis Force
We don’t have the time to go into detail on this fiction force, but youwill get a bonus point on the test if you can write a few sentences todescribe it, so do a web search on basic facts of the Coriolis force.
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 9 / 24
Linear and Angular physics
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 10 / 24
Linear and Angular physics
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 11 / 24
Linear and Angular physics
Everything we’ve learned so far translates almost easily into angularform. The hardest part is finding the equivalence of inertia. Let’srelate translational (or linear) kinetic energy to angular kinetic energyfor a particle going in a circle with radius R:
K =1
2mv2 =
1
2m(Rω)2 =
1
2mR2ω2 =
1
2
(mR2
)ω2
This tells us that for a single rotating particle, the equivalent inertia is
I = mR2
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 12 / 24
Moment of Inertia
In general (often there is more than one single particle), we will definethe angular equivlence of inertia (called the “Moment of Inertia”):
I =∑i
mir2i
For continuously distributed mass, such as in a solid body, we have:
I =
∫r2dm
where the integral is over the entire body.
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 13 / 24
Moment of Inertia
Find the moment of inertia for a thin rod about perpendicular linethrough center of rod?The rod has a total mass of M distributed evenly over a length L, sothe mass per unit length is M/L, and so
dm =M
Ldr
I =
∫ L/2
−L/2r2dm =
M
L
∫ L/2
−L/2r2dr =
M
3L
[L3
8+
L3
8
]=
1
12ML2
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 14 / 24
Parallel-Axis Theorem
According to this theorem, we can find the moment of inertia of a bodyaround any axis by relating it to a parallel axis that passes through thecenter of that body.
I = Icm + Mh2
Use this theorem to calculate the moment of inertia of a thin rod aboutan axis taken from one of its ends.
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 15 / 24
Newton’s Second Law (Rotation)
Now that we’ve achieved a definition for inertia in terms of rotation,we can use Newton’s Laws for scenarios involving rotation.Torque is defined to be τ = ~F~R, which is the rotational equivalent ofF = ma. Newton’s second law in rotation form is:∑
τext = Iα
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 17 / 24
Example 4: Non-Ideal Pulley
In previous pulley problems, wetreated the pulley as beingmassless. This is a reasonableapproximation whenever theweights involved are alwayssubstantially larger than the massof the pulley, but in general thepulley’s mass will have to beaccounted for. In this problem, awheel of radius 0.30 meters andmass 20 kg supports a 40 kg object.Find the acceleration of the blockin two scenarios: one in which weignore the pulley and another inwhich we do not. Treat the pulleyas a thin disk with I = MR2
4
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 18 / 24
The kinetic energy of a rolling wheel (non-slip) is:
K =1
2Icmω
2 +1
2Mv2
cm
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 21 / 24
Example 5
A ball of radius 0.50 meters andmass 10.0 kg roles down a ramp ofheight 5 meters. What is the finalspeed of the ball? Keep in mindthat “rolling friction” does not takeenergy out of the system, andassume that no slipping occurs (inwhich case energy would be lost).For a sphere, I = 1
3MR2.
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 22 / 24
Angular Momentum
L = r × P
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 23 / 24
Example 6
Find the total angular momentumof this system in terms of the givenvariables. The mass of the rim ofthe pulley is M, and you canconsider the rest of the pulley to bevirtually massless. Also find anexpression for the linearacceleration of the two masses.
Dr. Jones (Drexel) Physics 185F2013 Lecture Three November 5, 2013 24 / 24