# Stoke's Law air density calculation

Stokes' Law describing partial sedimentation is given by:

Rate = [(gd^2)(Δρ)]/18η

where g = 9.81 m s^-2 is gravitational acceleration. Δρ is the density difference between particles and the air, air viscosity of η = 1.67E-4 g cm^-1 s^-1 and d is the particle diameter in centimeters.

a. Calculate the weight(g) of one cubic centimeter of air at 1.0 atm and 15 degrees celsius.

b. Assume particles have a density of 2.45 g cm^-3 and are being released by a 100 m smokestack. How long does it take (s) a 25μm particle to settle to the ground?

c. It rains exactly once a week. What is the smallest particle size (μm) that can settle to the grounds before wet removal?

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#### Solution Preview

a. Calculate the weight(g) of one cubic centimeter of air at 1.0 atm and 15 degrees celsius.

if we use the equation alluded to in: http://hypertextbook.com/facts/2000/RachelChu.shtml (Center choice) as "The density of moist air may be determined by a similar relation:

D = 1.2929 (273.13/T) [(B - 0.3783e)/760] where T is the absolute temperature; B, the barometric pressure in mm, and e the vapor pressure of the moisture in the air in mm."

Then in our circumstance D = ...

#### Solution Summary

Solution to problem involving sedimentation/settling rates of airborne particles as per Stoke's Law.

Sound, Mass, Vector, Friction, Speed and Hooke's Law

1. According to a rule-of-thumb, every 3.0 seconds between a lightning flash and the following thunder gives the distance of the storm in km. Assuming that the flash of light arrives in essentially no time at all, estimate the speed of sound in kph from this rule.

2. A violent rainstorm dumps 1.0cm of rain on a city 4.4 km wide and 7.4 km long in a 2-h period. How many metric tons (1 ton = 103 kg) of water fell on the city? [1 cm3 of water has a mass of 1 gram = 10-3 kg.]

3. Vector V1 is 6.69 units long and points along the negative x-axis. Vector V2 is 2.84 units long and points at 45 degrees above the positive x-axis. What is the magnitude of the sum of these two vectors?

4. A fire hose held near the ground shoots water at a speed of 4.07 m/s. The water rises upwards in an arc then falls back to the ground. If the nozzle is pointed upwards at an angle of 29.31degrees, what distance away from the nozzle, in metres, does the water hit the ground?

5. A box is given a push so that it slides across the floor. How far will it go, in m, given that the coefficient of kinetic friction is 0.21 and the push imparts an initial speed of 3.8 m/s?

6. What is the minimum stopping distance for 119 kg car traveling 94 km/h if the coefficient of static friction between the tires and the road is 0.74? Give your answer in metres.

7. A car starts from rest and accelerates at a constant rate of 2.3 m/s2. If the radius of the car wheels is 28 cm, and the final angular speed of the car wheel is 63 rad/s, what distance (in m) does the car travel?

Hint: If you're unsure where to start, think about how you would find the final speed of the car first.

8. Calculate the velocity (in m/s) of a satellite moving in a stable circular orbit around the Earth at a height of 15,036 km above the Earth's surface.

Assume that the Earth's average diameter is 12,760 km and its mass is 5.97 x 1024 kg.

9. What is the minimum work needed to push a 1,021 kg car 122 m up a 17.7° incline? Ignore friction.

Give your answer in kJ where 1kJ = 1000J.

10. Devon wants to try shooting a stone from her slingshot straight into the air. If she stretches the rubber band of the slingshot back 25 cm from its relaxed position, and releases the stone from rest at this position, how high (in cm) will the 98 g stone travel with respect to its initial position (i.e. treat the initial height of the stone as zero before it is released)? Assume the rubber band obeys Hooke's Law with a spring constant of 262 N/m.

Neglect air resistance.

11. Karen has an average useful power output of 44.2 W. What would be the minimum time (in ms) it would take Karen to pull a spring a distance 55 cm beyond its relaxed position? Assume the spring constant has a value of 352 N/m.

12. A 2.6 m bar with negligible mass is fastened such that it hangs straight out from the side of a building. A cable is attached to the end of the metal bar on one end, with the other end attached to the building wall above the bar. The angle between the cable and the bar is 26.8 degrees (marked as theta in the image). If a 6.1 kg mass is hung from the bar at a distance of 2.4 m from the wall, what is the force of tension in the cable? Express your answer as a positive value in N, but do not include a '+' sign.

Hint: You will have to determine the two torques on the bar such that the system is in equilibrium. Set your pivot as the point where the bar is connected to the building. You should have one positive and one negative torque acting against one another.

13. A block of mass M = 1.3 kg is attached to a string that is wrapped around the circumference of a pulley of radius r = 5 cm and mass m = 0.71 kg. The pulley rotates freely about its axis and the string wraps around its circumference without slipping. The pulley rotates such that the mass rises with a linear speed of v = 0.94 m/s. What is the total kinetic energy of this system (pulley+mass)? The moment of inertia of the pulley is I = (1/2)mr2.

Hint: the linear speed of the block and the angular speed of the pulley are connected with the relation you are familiar with, i.e. angular speed = (linear speed divided by radius).

14. A force is exerted on the head of a nail which has a cross-sectional area of 3.2 cm2. The resulting pressure on the head of the nail is 339 kPa. What is the magnitude of the force needed to exert such a pressure? Give your answer in N.

15. Assume you are given two objects whose centre of mass is located exactly at the origin. The first object has a mass of 33 kg and is located at position +30 m with respect to the origin. If the mass of the second object is 7.0 kg, what is its position (in m) with respect to the origin (be specific with the sign of the position).

16. Two cars leave the same point at the same time, each travelling at the same constant speed of 101 km/h and each having the same mass. However, the first car drives directly east, while the second car heads directly south. At what speed (the magnitude of the velocity) is the centre of mass of these two cars travelling? Give your answer as a positive value in km/h.

17. A gun is fired horizontally under water. The bullet has a spherical shape of radius 0.66 mm and a mass of 3.1 g. The temperature of the water is 20°C. If its current speed is 88 m/s, calculate the magnitude of its current acceleration in the x-direction in m/s2 (give a positive answer with no sign). Assume the bullet travels horizontally, i.e. neglect gravity and the buoyant force. The viscosity of water at 20°C is 1.0 x 10-3 Pa s.

18. A garden hose is used to fill a large metal container which can hold 21 L of liquid. If the radius of the garden hose's nozzle is 1.1 cm, and the speed of the water at this point is 273 cm/s, how long (in seconds) would it take to fill the container with water, assuming that none of it splashes out?

Hint: Use the fact that 1 L = 1000 cm3

19. A sound wave has an intensity of 37 mW/m2 and interferes destructively with another sound wave. The result (assuming complete destructive interference) is an intensity of 18 mW/m2. What would the intensity of the second sound wave have to have been in order to produce such an intensity? Give your answer in mW/m2.

Problem 1. An object is thrown at initial speed vo = 30 m/s , at the elevation angle θ = 30 from

a H=65 m high building.

Neglecting the air resistance calculate:

a) the time of flight;

b) the maximal altitude of the object;

c) the distance of the object, when it hits the ground, measured from its initial position;

d) eliminate time from x(t) and y(t) and draw an approximate diagram y(x) (trajectory) of the

object.

Problem 2. An object of mass m= 5.00 kg starts moving , at initial speed vo= 1.5 m/s, from the

top of a smooth incline, θ = 30o.

The object is initially h=2.00 m higher then the end of an elastic spring, k= 60 N/cm.

a) calculate the maximal speed of the object ;

b) calculate the maximal compression of the spring;

c) briefly describe what will happen after.

Problem 3. a) Calculate the approximate distance between the Earth and the Sun assuming

that the mass of the Sun is 2x1030 kg, G = 6.67x 10 -11 Nm2/kg2 , that the period is 365 days,

and that the orbit is circular.

b) calculate the average speed of the Earth with respect to the Sun;

c) calculate the radial acceleration of the Earth.

d) calculate the gravitational potential energy assuming that the mass of the Earth is 6x 10 24 kg.