where Mis the mass of the cart (mass of water moving with cart is negligible)
From conservation of mass
Combining terms
Since the jet velocity is constant
138
6.95: PROBLEM DEFINITION
Situation:
A problem in rocket-trajectory analysis is described in the problem statement.
Find:
Initial mass of a rocket needed to place the rocket in orbit.
SOLUTION
139
6.96: PROBLEM DEFINITION
Situation:
A toy rocket is powered by a jet of water–additional details are provided in the
problem statement.
Find:
Maximum velocity of the rocket.
Assumptions:
Neglect hydrostatic pressure; Inlet kinetic pressure is negligible.
SOLUTION
Newtons 2nd law. XF=ma
where T=thrust and W=weight
where vR=0when t=0.Then
The exit pressure is zero (gage) and the inlet kinetic pressure is negligible. So
140
Timeforthewatertoexhaust:
Thus
141
6.97: PROBLEM DEFINITION
Situation:
Water is discharged from a slot in a pipe–additional details are provided in the
problem statement.
Find:
Reaction (Force and Moment) at station A−A
PLAN
Apply the momentum equation and the moment of momentum equation.
SOLUTION
Momentum equation (y-direction)
Flow rate
Momentum equation (z–direction)
142
Moment-of-momentum (z−direction)
Moment-of-momentum (y-direction)
143
6.98: PROBLEM DEFINITION
Situation:
A helicopter rotor uses two small rockets motors–details are provided in the prob-
lem statement.
Find:
Power provided by rocket motors.
PLAN
Apply the momentum equation. Select a control volume that encloses one motor,
and select a stationary reference frame.
SOLUTION
Velocity analysis
Flow rate
Momentum equation (x-direction)
Power
6.99: PROBLEM DEFINITION
Situation:
A rotating lawn sprinkler is to be designed.
The design target is 0.25 in. of water per hour over a circle of 50-ft radius.
Find:
Determine the basic dimensions of the lawn sprinkler.
Assumptions:
The Bernoulli equation applies.
Assume mechanical friction is present.
PLAN
Apply the momentum equation.
SOLUTION
Flow rate. To supply water to a circle 50 ft. in diameter at a 1/4 inch per hour
requires a discharge of
If the water were to exit the sprinkler head at the angle for the optimum trajectory
(45o), the distance traveled by the water would be
145
atomization will occur which produces droplets. These droplets will experience aero-
dynamic drag which will reduce the distance of the trajectory. The size distribution
of droplets will lead to small droplets moving shorted distances and larger droplets
farther which will contribute to a uniform spray pattern.
The sprinkler head can be set in motion by having the water exit at an angle with
respect to the radius. For example if the arm of the sprinkler is 4 inches and the
angle of deflection at the end of the arm is 10 degrees, the torque produced is
The moment necessary to overcome friction on a flat plate rotating on another flat
plate is
M=(2/3)μFnro
where μis the coefficient of friction and rois the radius of the plate. Assuming a 1/2
inch radius, the limiting coefficient of friction would be
146
6.100: PROBLEM DEFINITION
Situation:
Water flows out a pipe with two exit nozzles–additional details are provided in
the problem statement.
30
o
1
2
3
Find:
Reaction (Force and Moment) at section 1.
PLAN
Apply the continuity equation, then the momentum equation and the moment of
momentum equation.
SOLUTION
Continuity equation equation
Momentum equation (x-direction)
Momentum equation (y-direction)
Moment-of-momentum (z−direction)
6.101: PROBLEM DEFINITION
Situation:
Water flows out a pipe with two exit nozzles–additional details are provided in
the problem statement.
2y
x
3
30
o
Find:
Reaction (Force and Moment) at section 1.
PLAN
Apply the continuity equation, then the momentum equation and the moment of
momentum equation.
SOLUTION
Continuity equation equation
Momentum equation (x-direction)
Momentum equation (y-direction)
Weight
148
thus
Moment-of-momentum (z−direction)
Reaction at section 1
149
6.102: PROBLEM DEFINITION
Situation:
A reducing pipe bend held in place by a pedestal. Water flow. No force trans-
mission through the pipe at sections 1 and 2.
Assume irrotational flow. Neglect weight
Find:
(a) Force needed to hold bend stationary: F
(b)Momentneededtoholdbendstationary: M
PLAN
Apply the Bernoulli equation, then the momentum equation, and then the moment
of momentum equation.
SOLUTION
Bernoulli equation
Momentum equation (x-direction)
where
thus
Moment-of-momentum (y-direction)
Net force and moment at 3
151
6.103: PROBLEM DEFINITION
Situation:
Centrifugal fan is used to pump air
Find:
Power (hp) required to operate fan
Assumptions:
Neglect the compressibility of air.
PLAN
Apply the moment of momentum equation between inlet and outlet.
SOLUTION
The control volume enclosed the rotor but does not rotate. The flow is steady within
the control volume. Assume positive direction comes out of the page, the ezdirection.
The moment diagram shows one moment (torque)
Since the flow is steady and there is not inflow of moment of moment, the equation
reduces to
152
The exit radial velocity is
At the outlet
(r×v)o=D
2ωD
2ez
Thetorqueis
The power is
153