Problems: include all the details of your calculation and circle your final answer.

Problem 1

A crossbow with a mass of 30.0 kg is used to shoot a bolt whose mass is 0.050 Kg. Thecrossbow and the bolt are initially at rest.

a) If a bolt leaves the crossbow with a speed of 100 m/s, what is the recoil speed of thecrossbow?

b) In which directions are the bolt and the crossbow moving?

c) Compute the initial and final kinetic energy of the bolt and the crossbow.

Problem 2

A block of wood is attached to the bottom of a lake by means of a rope and it is completelyimmersed in the water. The wood has a density of 0.25 × 103kg/m3 (while water has adensity of 1.0× 103kg/m3).

a) If the the Volume of the block is V = 0.0136m3, what is the tension in the string?

b) If we cut the rope, the block will start accelerating towards the surface. Calculate theacceleration.

Problem 3

A block of wood whose volume is 0.01 m3 is attached to the bottom of a lake by means of arope and it is completely immersed in the water. The wood had a density of 0.25×103kg/m3

(while water has a density of 1.0× 103kg/m3).What is the tension in the rope?

Problem 4

The pressure in a cylinder filled with air is 1 atm when the volume is 0.03m3. A temperatureT = 27◦C is kept constant.

a) We compress the gas without changing the temperature until the pressure doubles.What is the final volume?

b) How many moles of air do we have in the cylinder?

Problem 5

A thermal engine performs 200 J of mechanical work and discards 300 J of heat in eachcycle. How much energy should we provide to the engine (as heat) in each cycle? What isthe efficiency of the engine?

Problem 6

During each cycle, a thermal engine absorbs a quantity of heat of QH = 1000 J and discardsQC = 750 J. What is the efficiency of the engine? How much work will be produced in eachcycle?

Problem 7

A swimming pool has a depth of 5.0 meter and it is filled with water (ρ = 1.0× 103 kg/m3).The atmospheric pressure is patm = 1.013× 105 Pa.Calculate:

a) The pressure 1.0 meters below the surface of the water.

b) The pressure at the bottom of the pool.

c) The buoyant force acting on a block of volume 0.5 m3 located 3.0 meters below thesurface of the water.

d) What is the buoyant force acting on the same block if we place it at the bottom of thepool?

Problem 8

A fisherman counts the number of water waves that pass under his boat. He counts 15 wavesin one minute. He also measures that the displacement of his boat from the highest to thelowest position is 1.2 meters and that the distance between two consecutive wave crests is 3.0meters. What are the frequency, the wavelenght, the period, the speed and the amplitudeof the waves?

Newton’s Law:

~F = m~a

Friction force (kinetic):

f = µkN

Universal law of gravitation:

F = Gm1m2

d2

G = 6.67× 10−11Nm2/Kg2

Satellite motion:

v =

√

GME

r

Circular motion (radius R and period T )

v =2πR

T

a =v2

R

Mechanical Energy:

K =1

2mv2

U = mgh

E = K + U

Conservation of energy:Einit = Efinal

Work:W = Fd cosα

Work-energy theorem:Einit +W = Efinal

Power:P = W/(∆t)

Definition of momentum:~p = m~v

Conservation of momentum:~pinitial = ~pfinal

m1v1,i +m2v2,i = m1v1,f +m2v2,f

Waves:v = λf , f = 1/T

Pascal’s law:p = pa + ρgh

Buoyant force:B = ρfluidV g

Density:

ρ =m

V

Heat:Q = mc∆T , Q = mL

Temperature conversion (Kelvin-Celsius):

TK = TC + 273.15

Pressure Conversion (atm-Pa):

1 atm = 1.013× 105Pa

Equation of Ideal Gas:pV = nRT R = 8.3145 J/(mol◦K)

First Law of Thermodynamics:∆U = Q−W

W = p∆V

Efficiency of a thermal engine:

η =W

QH

= 1−QC

QH