Dr. Galeazzi PHY205 Test #2 October 12, 2011 Signature ...rads.physics.miami.edu/optics/ken/Class205_F2011/Exams_F2011/Midterm2... · Dr. Galeazzi PHY205 Test #2 October 12, 2011

Dr. Galeazzi PHY205 Test #2 October 12, 2011

1

Signature: ___________________________ Name:_______________________________ I.D. number: _________________________ You must do the first problem which consists of five multiple choice questions. Then you must do three of the four long problems numbered 2-5. Clearly cross out the page and the numbered box of the problem omitted. Do not write in the other boxes. If you do all the problems, only problems 1-4 will be graded. Each problem is worth 25 points for a total of 100 points. TO GET CREDIT IN PROBLEMS 2 – 5 YOU MUST SHOW GOOD WORK. CHECK DISCUSSION SECTION ATTENDED: [ ] Dr. Ghandour 1O, 9:30 – 10:20 a.m. [ ] Dr. Ghandour 1P, 11:00 – 11:50 a.m.

[ ] Dr. Zuo 1Q, 12:30 – 1:20 p.m. [ ] Dr. Mariano 1R, 2:00 – 2:50 p.m.

[ ] Dr. Mariano 1S, 3:30 – 4:20 p.m. [ ] Dr. Cohn 1T, 5:00 – 5:50 p.m.

[ ] Dr. Zuo 2O, 9:30 – 10:20 a.m. [ ] Dr. Cohn 2Q, 12:30 – 1:20 p.m.

[ ] Dr. Nepomechie 2T, 5:00 – 5:50 p.m.

EQUATION SHEET

A detachable equation sheet is provided on the last page of the test

1

2

3

4

5

TOTAL


2

[1.] This problem has five multiple choice questions. Circle the best answer in each case. [1A.] When an object moves very fast in air, the air resistance is proportional to the square of

the speed and it is called the drag force: 𝑓 = 𝐷𝑣!, where 𝐷 is constant. Assume that a person with mass m jumps from an airplane flying at an altitude h and is subject to the weight and drag only. What is her terminal speed?

[a] !"!

[b] 2𝑔ℎ [c] 0 [d] 2𝑔ℎ [e] !"!

[1B.] A mass 𝑀 is initially at rest on an incline that forms an angle 𝛼 with the horizontal. The

net force on the mass is zero and the mass stays at rest. What is the magnitude of the static friction between the mass and the incline?

[a] 𝜇!𝑀𝑔 sin𝛼 [b] 𝜇!𝑀𝑔 cos𝛼 [c] 0 [d] 𝑀𝑔 cos𝛼 [e] 𝑀𝑔 sin𝛼

[1C.] A mass 𝑀 is initially sliding down an incline that forms an angle 𝛼 with the horizontal.

The net force on the mass is zero and the mass slides down at constant speed 𝑣!. What is then the magnitude of the static friction between the mass and the incline?

[a] 𝜇!𝑀𝑔 sin𝛼 [b] 𝜇!𝑀𝑔 cos𝛼 [c] 0 [d] 𝑀𝑔 cos𝛼 [e] 𝑀𝑔 sin𝛼

[1D.] What should be the acceleration of an elevator that makes you feel weightless? Assume

that the positive y-axis points up.

[a] +2𝑔𝚥 [b] −2𝑔𝚥 [c] 0 [d] +𝑔𝚥 [e] −𝑔𝚥 [1E.] You are crossing a river which moves at speed 𝑣! with respect to the bank, with a boat

that moves at speed 𝑣! with respect to the river. If you point your boat straight across the river (as seen from the water), what is the angle between the direction of motion of the boat and the direction straight across the river, as seen by an observer on the river banks.

[a] 0 [b] sin!! !!

!! [c] tan!! !!

!! [d] sin!! !!

!! [e] tan!! !!

!!


3


4

[2.] A mass 𝑀 is initially held at rest at the top of an incline of length L, with slope forming an angle 𝛼 with the horizontal (see figure). When the mass is released, it starts sliding down the incline. The coefficient of kinetic friction between the mass and the incline is 𝜇!.

a) Draw a free body diagram for the mass. b) Find the acceleration of the mass.

c) Find the time it take the mass to reach the bottom of the incline.

Write your results in terms of M, L, 𝛼, 𝜇!, and g. Check the units/dimensions for each answer.

______________________________________________________________________________


5


6

[3.] The velocity of a particle as a function of time is given by:

𝑣 = 2𝑚𝑠 𝚤 + 12

𝑚𝑠! 𝑡! − 6

𝑚𝑠! 𝑡 𝚥

With 𝑟 𝑡 = 0 = 5𝑚 𝚥.

a) Find the position of the particle as a function of time, 𝑟 𝑡 .

b) Find the trajectory of the particle, 𝑦(𝑥).

c) Find the time(s) when the net force on the particle is zero.

______________________________________________________________________________


7


8

[4.] In the system shown in the figure, a mass M is connected to two springs of constants 𝑘! and 𝑘! respectively. The springs are in their unstretched position when the mass is in equilibrium and there is no friction. If the mass is moved from equilibrium, it starts moving in simple harmonic motion. a) Show that, when the mass is at a distance x from equilibrium, the net force on the mass is

𝐹! = − 𝑘! + 𝑘! 𝑥. (be careful with the directions of the individual forces and make sure you explain what you are doing!)

b) Starting from the “definition” of simple harmonic motion, derive the acceleration 𝑎! of the mass M, and express it as a function of the distance x from equilibrium.

c) Using the results in (a) and (b), find the period of the simple harmonic motion.

Write your results in terms of M, 𝑘!, 𝑘!, and 𝜔. Check the units/dimensions for each answer.

______________________________________________________________________________


9


10

[5.] A new park attraction consists of a vertical wheel of radius R, spinning at constant angular velocity, with cars for passengers. The passengers feel weightless at the top of the wheel. Assume that the typical mass of a passenger is 𝑚.

a) What is the magnitude of the force exerted on a passenger by the car at the top of the wheel?

b) What is the speed of the car? c) What is the magnitude of the force exerted on a passenger by the car at the bottom of

the wheel? d) What is the magnitude of the force exerted on a passenger by the car when they are 90

degrees from the bottom, going up? Write your results in terms of R, m, and g. Check the units/dimensions for each answer. ______________________________________________________________________________


11


12

Equation Sheet

Vectors: 2D: 𝐴 = 𝐴!𝚤 + 𝐴!𝚥, 𝐴 = 𝐴 = 𝐴!! + 𝐴!! ; 𝜃 = tan!! !!!!

; 𝐴! = 𝐴 cos 𝜃 ,𝐴! = 𝐴 sin 𝜃.

𝐴 ∙ 𝐵 = 𝐴𝐵 cos𝜙 = 𝐴!𝐵! + 𝐴!𝐵! + 𝐴!𝐵!; 𝐴×𝐵 = 𝐴𝐵 sin𝜙 + R.H.R. for direction

Linear motion: 𝑟 = 𝑥𝚤 + 𝑦𝚥 + 𝑧𝑘; 𝑣 = !!!"= 𝑣!𝚤 + 𝑣!𝚥 + 𝑣!𝑘; 𝑎 =

!!!"= 𝑎!𝚤 + 𝑎!𝚥 + 𝑎!𝑘

𝑥! − 𝑥! = 𝑣!𝑑𝑡; 𝑣!! − 𝑣!! = 𝑎!𝑑𝑡!!!!

!!!!

; 𝑣!" =!!!!!!!!!!

; 𝑎!" =!!!!!!!!!!

Relative velocity: 𝑣!/! = 𝑣!/! + 𝑣!/!; 𝑣!/! = −𝑣!/!

Newton’s Laws: 𝐹 = 0 ⟺ 𝑎 = 0; 𝐹 = 𝑚𝑎 = !!!"

; 𝐹!" = −𝐹!"

Examples of forces: gravity: 𝐹! = 𝑚𝑔; 𝐹! = 𝐺 !!!!!!

; spring: 𝐹! = −𝑘𝑥; friction: 𝑓! = 𝜇!|𝑛|,

0 < 𝑓! = 𝜇!|𝑛|,

Circular Motion: 𝜔 = !!!";𝛼 = !!

!"

Uniform circular motion: 𝑟 = 𝑅 𝑐𝑜𝑠𝜃𝚤 + 𝑠𝑖𝑛𝜃𝚥 = 𝑅𝑟; 𝑣 = 𝜔𝑅 −𝑠𝑖𝑛𝜃𝚤 + 𝑐𝑜𝑠𝜃𝚥 = 𝜔𝑅𝜃;

Non-uniform circular motion: 𝑎!"! = 𝑎!"# + 𝑎!"#; 𝑎!"# = 𝜔!𝑅; 𝑎!"# = !"!"= 𝑅𝛼

Periodic motion: 𝑇 = !!; 𝜔 = 2𝜋𝑓; SHM: 𝑥 = 𝐴 cos 𝜔𝑡 + 𝜙 ;

Dr. Galeazzi PHY205 Test #2 October 12, 2011 Signature ...rads.physics.miami.edu/optics/ken/Class205_F2011/Exams_F2011/Midterm2... · Dr. Galeazzi PHY205 Test #2 October 12, 2011

Documents