Mechanics | Relativity Answers

Question Five
(a) (i) Solve the following simultaneous equation using Cramer’s rule
x + y + z = 4
2x – 3y + 4y = 33
3x – 2y – 2z = 2
(ii) Find the eigen values and their corresponding eigen vectors for the matrix

A garden hose attached with a nozzle is used to fill a 10-gal bucket. The inner diameter of the hose is 2 cm, and it reduces to 0.8 cm at the nozzle exit. density of water 1000 kg/m3; 1 gal = 3.789 x 10-3 m3.

If it takes 50 s to fill the bucket with water, determine the volume flow rates of water through the hose, and the average velocity of water at the nozzle exit.

A steady, incompressible, two-dimensional flow field defined by the velocity vector is expressed as

V = (u,v) = (0.5 + 0.8x)i + (1.5 - 0.8y)j

where the x- and y-coordinates are in meters and the magnitude of velocity is in m/s.

Calculate the material acceleration at the point (x = 2 m and y = 4 m)


Select one:
a. ax = 1.68 m/s2; ay = 1.36 m/s2

b. ax = 0.72 m/s2; ay = 1.68 m/s2
c. ax = 1.68 m/s2; ay=0.72 m/s2
d. ax = 2.45 m/s2; ay = 1.72 m/s2

The aerodynamic drag of a new sports car is to be predicted at a speed
of 50.0 mi/h at an air temperature of 25 °C. Automotive engineers build
a one-fifth scale model of the car to test in a wind tunnel. It is winter and
the wind tunnel is located in an unheated building; the temperature of
the wind tunnel air is only about 5 °C. Determine how fast the engineers
should run the wind tunnel in order to achieve similarity between the
model and the prototype. Properties of air at 25 °C are density 1.184
kg/m3 and dynamic viscosity is 1.849 * 10-5
, at 5 °C density is 1.269
kg/m3 and dynamic viscosity 1.754 * 10-5.

A 6-cm diameter horizontal pipe expands gradually to a 9-cm
diameter pipe (Fig below). The walls of the expansion section are
angled 10° from the axis. The average velocity and pressure of
water before the expansion section are 7 m/s and 150 kPa,
respectively. Determine the head loss in the expansion section an
the pressure in the larger-diameter pipe.
Properties: density of water = 1000 kg/m3
, total angle inclination
20°, diameter ratio d/D = 6/9, KL = 0.133.

Consider the fully developed flow of glycerin at 40 °C through a 70-mlong, 4-cm-diameter, horizontal, circular pipe. If the flow velocity at the
centerline is measured to be 6 m/s, determine the velocity profile and the
pressure difference across this 70-m-long section of the pipe, and the
useful pumping power required to maintain this flow.

 How many regimes of fluid
flow do you know?
 What are the these regimes
called?

Water is accelerated by a nozzle to an average speed of 20 m/s, and strikes
a stationary vertical plate at a rate of 10 kg/s with a normal velocity of 20
m/s (Fig. below). After the strike, the water stream splatters off in all
directions in the plane of the plate. Determine the force needed to prevent
the plate from moving horizontally due to the water stream.

A fan is to be selected to cool a computer case whose dimensions are 12 cm
3 40 cm x 40 cm (Fig. below). Half of the volume in the case is expected to
be filled with components and the other half to be air space. A 5-cmdiameter hole is available at the back of the case for the installation of the
fan that is to replace the air in the void spaces of the case once every
second. Small low-power fan–motor combined units are available in the
market and their efficiency is estimated to be 30 percent. Determine (a) the
wattage of the fan–motor unit to be purchased and (b) the pressure
difference across the fan. Take the air density to be 1.20 kg/m3.

1. Consider the steady adiabatic flow of an incompressible fluid. Can
the temperature of the fluid decrease during flow? Explain.
2. Consider the steady adiabatic flow of an incompressible fluid. If the
temperature of the fluid remains constant during flow, is it accurate
to say the frictional effects are negligible?
3. A person is filling a knee-high bucket with using a garden hose and
holding it such that water discharges from the hose at the level of his
waist. Someone suggests that the bucket will fill faster if the hose is
lowered such that the water discharges from the hose at knee level.
Do you agree with this suggestion? Explain. Disregard any frictional
effects.