Physics 201, Review 2

Important Notes: v  This review does not replace your own preparation efforts v  The review is not meant to be complete. v  Exercises used in this review do not form a test problem pool. Please practice more with end of chapter problems.

About Exam 2 q  When and where

§  Monday Oct 23rd, 5:30-7:00 pm §  125 Ag Hall.

q  Format §  Closed book §  One 8x11 formula sheet allowed, must be self prepared. (Absolutely no photocopying/printing of sample problems,

examples, class lectures, HW etc.) §  20 multiple choice questions. §  Bring a calculator (but no computer). Only basic calculation

functionality can be used. q  Special needs/ conflicts:

§  All requests for alternative test arrangements should have been settled by now. (except for medical emergency)

§  All alternative test sessions are in our lab room, only for approved requests.

Chapters Covered q  Chapter 5: Laws of Motion q  Chapter 6: Circular Motion and Application of Newton’s Laws

q  Chapter 7: Energy (Section 7.9: Energy diagram/equilibrium postponed till Chapt12/midterm3)

q  Chapter 8: Conservation of Energy

Reminders: §  Tomorrow is another Super Friday! (10am-5pm, in the lab room) §  The current homework assignment HW 7 is due next Wednesday.

Please try to work it out before the exam though.

Basic Concepts and Quantities q  Forces:

§  Common forces: Weight, Normal Force, Friction (static, kinetic), Tension, spring force.

§  Action Reaction §  Free Body Diagram

q  Dynamics: §  F=ma

q  Non-Inertia Reference Frame: Fictitious Force (conceptual only) q  Energy, Work.

§  Kinetic Energy §  Potential Energy : definition, gravitational, elastic. §  Work: definition, conservative, non-conservative. §  Energy Conservation. §  Power

q  Application specific: apparent weight, centripetal force.

Basic Techniques q  Identify forces. Draw Free Body diagrams q  Decompose a vector quantity (v, a, F) into x, y projection. q  Set up equations for Newton’s Law in each projection. q  Calculate kinetic energy. q  Calculate work

§  done by general force (simple math) §  via Work-Energy relationship §  by a conservative force (weight, spring force).

q  Calculate potential energy for weight/gravity, spring q  Set up basic equations based on energy/work relationship q  Deal with applications such as:

§  Pulley and masses; Mass on a Slope §  Static and Kinetic Friction §  Car turning on horizontal (or banked) curve. §  Moving on slope, hill, curved track (energy approach)

Review: Dynamics (Force and Newton’s Laws)

q  Common forces: §  Weight Fg=mg pointing downwards §  Normal N always normal to the surface pointing towards object §  Friction:

•  kinetic: fk= µkN , along surface opposite to motion •  static: fs ≤ µsN , along surface opposite to motion trend

§  Tension: same magnitude along string, always in string direction away from object.

§  Spring Force Fs =-kx (Hooke’s Law) q  Newton’s 2nd Law:

q  Newton’s 3rd Law: F12= - F21

amFF i

==∑

22

ωmrrvmmaF

maFmaFmaF

cc

zz

yy

xx

===

=

=

=

A “Trivial” Conceptual Quiz q  A block of mass m is being dragged at a constant velocity by a

horizontal force F on a horizontal surface with friction. (See fig). Ø  Which of the following force pair is action-reaction pair? §  A: F and fk, B: N and mg, C: both of above, D: none of above.

F fk

N

mg

Key: Newton’s 3rd Law is about two forces on two objects.

Reminder: conceptual problems will make 1/3 of the exam.

Another “Trivial” Conceptual Quiz q  A As shown below, an ideal spring with Hooke’s constant of

k=0.3N/m is being compressed at both ends by two forces of equal magnitude. If the magnitude of each force is 0.3N, what is the compression (i.e. the difference between the uncompressed and the compressed length) ?

§  A: 0m, B: 1.0m, C: 2.0m , D: Some other value.

Problem Solving Procedure q  Identify all objects of interest in the problem q  Draw Free Body Diagram for each of those objects, if forces are

involved. q  Set up x,y,z axes for each object. The orientation of the axes can be

different for different objects. Choose the ones that fit best for the particular setting in the problem.

q  Decompose vector variables on axes. q  Write down equations based on Newton’s Laws and/or kinematics,

separately for each object, in each direction. q  Explore relationship between parameters among objects. (e.g. objects

connected by tight string will have same acceleration value) q  Solve equations for all unknowns. q  Try to do all above analytically if possible. Use your calculator only at

this stage to get numerical values. q  Make sure your answers are not too ridiculer. (A car runs at 1000m/s?) q  Supply units to all numerical answers . q  Double check and submit your answer.

A “Trivial” Exercise: Throwing Baseball q  The gravitational force exerted on a baseball is 2.24 N down. A

pitcher throws the ball horizontally with velocity 14.0 m/s by uniformly accelerating it along a straight horizontal line for a time interval of 187 ms. The ball starts from rest.

What are the magnitude of the force the pitcher exerts on the ball? q  Solution A: §  mg = 2.24 N m= 2.24/9.8 = 0.229 kg §  v=v0 + at a = (14-0)/0.187 = 74.87 m/s2 §  F = ma = 0.23 x 74.9 = 17.1N

Correct ? No! q  You need to DRAW FBD FIRST. (Workout the correct solution after the class, this was in HW 4-5)

N Fs

Exercise 2: Block On a Slope q  The static coefficient of friction between the block and the

slope surface is µs=0.3, what is the maximum θ before the block start to slide?

A: 10o , B: 17o , C: 23o , D: 42o E: 90o q  Solution: §  Step 1: Draw free body diagram §  Step 2: Choose x,y coordinator system, decompose forces §  in y: N - mgcosθ =0 N = mgcosθ §  in x: -Fs + mgsinθ =0 Fs = mgsinθ < µsN = µsmgcosθ tanθ< µs θ < 16.7o

Exercise 3: Block Inside An Accelerating Car q  A block of 5kg is sitting on a frictionless floor of a train car that

is accelerating at a= 6.0 m/s2 .The block is attached on a light cable with a light spring scale as shown. There is no relative motion between the block and the car. What is the reading of the scale? §  A: 15 N §  B: 30 N §  C: 60 N §  D: 49 N §  E 0 N

a=6.0 m/s2

Block moves together with the car: a=6.0 m/s2 horizontally T=ma =5x6=30N (Scale reads T, not 2T)

x

y

N

mg

T

Note: One can also solve this problem by using concept of fictitious force in accelerating frame. However, please note fictitious force is not a real force, it is only a convenient tool.

Exercise 4: Motion with a Pulley q  Two objects of 3.0 kg and 5.0kg are connected by a soft string

around a frictionless pulley as shown. §  What is the tension on the string? (g=9.8 m/s2) A: 15.2 N, B: 29.4 N, C: 49N, D: 36.8N Solution: Ø  Free Body Diagrams Ø  Coordinate system Ø  Equations:

§  Object 1: T - M1g = M1a1

§  Object 2: M2g – T = M2a2

Ø  Connection: a1=a2=a Ø  Solve:

T=2M1M2/(M1+M2)g = 36.75 N More question to ask: What are the acceleration of the objects, respectively?

M1g

T

M2g

T

+x

+x

Review

Review Exercises

Exercise 5: Geostationary Satellite q  Geosynchronous satellites orbit the Earth at a distance of

42,000 km from the Earth's center. Their angular speed at this height is the same as the rotation rate of the Earth, so they appear stationary at certain locations in the sky. What is the force acting on a 1,500-kg satellite at this height? §  A: 457N §  B: 85N §  C: 333N §  D: 404N §  E: 514N

F=mac = m rω2

= 1500 x 4.2x107 x( 2π/86400)2

= 333N

Additional exercises: Try some car turning exercises yourself. (See lecture 10)

Reminder: Energy and Work q  Kinetic Energy: KE ≡ ½ mv2 q  Potential Energy: always associated with a conservative force

§  for weight : Umg = mgh §  for spring : Uelastic = ½ kx2

q  Work: generally

§  by conservative force: W = - (Uf –Ui) •  = -mg(hf-hi) by weight •  = -( ½ kxf

2 – ½ kxi2) by spring force

q  Energy Conservation, two forms:

§  KEf – KEi = Σ W (total work) KEf = KEi if no work at all. §  (KEf +PEf) - (KEi +PEi) = ΣWnon_conservative KEf +PEf = KEi +PEi if ΣWnon_conservative = 0

on a segment : W ≡F •Δx = F cosθ Δx

on a path: W ≡F •dx

path∫

Exercise 6 : Conceptual Quiz q  A block is being pushed up along an inclined plane;

A.  Work done by gravity is positive, done by kinetic friction is negative B.  Work done by gravity is negative, done by kinetic friction is positive C.  Both are positive D.  Both are negative.

v  Work done by gravity = -mgΔh < 0 if Δh>0; v  Work done by kinetic friction (on a stationary surface) is always negative

One additional quiz: A book of mass m, originally resting on the ground, is being picked up and put rest on a table of height h. What is the work to the book done by the picking up force? (0, mgh, -mgh, can not be determined)

A Conceptual Quiz with Circular Motion q  A particle of mass M, attached by a string with one fixed end at

center C, is undergoing a uniform circular motion on a smooth horizontal table. (See figure below for a top view). The length of the string is R. The period of particle’s motion is T. Be convinced that to maintain the uniform motion, the tension on the string must equal to FT=MR(2π/T)2.

Ø  During a half-period motion ( say, from point B to A), what is the work done by the tension? Α.  πRFT B.  2RFT C.  -2RFT D.  zero E.  none of above.

One More Conceptual Quiz q  Two mass blocks of 3.0 kg and 5.0kg are connected by a soft string

around a frictionless pulley as shown. As the masses are moving, how does the system’s potential energy change?

A: Increasing B: Decreasing C: remains unchanged D: I would need more time to do calculation E: Not enough information to determine as the friction constants are

not given

Problem Solving Procedure Ø  Identify system (moving objects + “earth” + “spring”) Ø  Identify all forces (remember Free Body Diagram?)

Ø  For each conservative force, identify/use potential energy Ø  For each non conservative force, calculate work(W).

Ø  Identify mechanic energy for initial and final states (Ei and Ef) Ø Ei = KEi +PEi = ½ mvi

2 + mghi + ½ kxi2 (if any)

Ø Ef = KEf +PEf = ½ mvf2 + mghf + ½ kxf

2 (if any)

Ø  Use energy conservation:

W = ΔE = Ef – Ei ( if W=0, use Ef = Ei )

Ø  Solve for unknowns

Exercise 7: Climbing up a Mountain q  What is the minimum amount of energy required for an 80-kg

climber carrying a 20-kg pack to climb Mt. Everest, 8 850 m high? (g=9.8 m/s2 )

A: 8.67 MJ, B:4.16 MJ, C:2.47 MJ, D:1.00 MJ

At bottom KEi=0, PEi=0 ; At top KEf=0, PEf=mgh Minimum increase of mechanic energy is the increase of PE: ΔPE = mgh = (80+20)x9.8x8850 = 8.67 MJ

Exercise 8: Power q  A car of a total mass of 1200 kg (including passengers) is

climbing up a slope of 18o at a constant speed. If the maximum power its engine can provide is 18.2 kW, what is the maximum speed at which the car can maintain on the slope?

Ø  3.5 m/s , 4.5 m/s, 5.0 m/s, other

P = ΔEΔt

ΔE=ΔKE+ΔPE=0 + mgΔh=mgΔxsinθ

→ P = mgΔxsinθΔt

=mgvsinθ (note: ΔxΔt

= v)

→ v = pmgsinθ

=18.2x103

1200x9.8xsin18o = 5.0 m/s

Exercise 9: Smooth Inclined Plane q  A block M = 1kg is pulled up along an inclined smooth plane

by a constant force F= 20N. The block starts from rest at the bottom, what is its speed when it reaches the top?

A: 5.1 m/s, B: 6.4 m/s, C: 7.8 m/s, D: 12.5 m/s

θ=30ο

F

Ef = KEf +PEf =½ mv2 + mgh Ei = (KEi +PEi )=(0+0) W = FΔs = Fd W = Ef – Ei ½ mv2 + mgh – (0+0) = Fd à ½ mv2 = Fd- mgh = Fd –mg d sin(30o) à  v = 7.77 m/s

h

Exercise 10: Inclined Plane w/ Friction q  A block M = 1kg is pulled up along an inclined plane by a

constant force F= 20N. In between the block and the plane surface µk=0.1. The block starts from rest at the bottom, what is its speed when it reaches the top?

Slightly more involved but very similar to the previous exercise. See board for an illustration and fill up details yourself.

θ=30ο

F

h