6.83: PROBLEM DEFINITION
Situation:
Lift and drag forces are being measured on an airfoil that is situated in a wind
tunnel–additional details are provided in the problem statement.
yp
u
8m
/
s
2
12 m
/
s0.25 m
0.25 m
L
D
0.5 m
xp
l
1m
c.v. c.s.
1
10 m/s
Find:
(a)Liftforce:L
(b) Drag force: D
PLAN
Apply the momentum equation.
SOLUTION
Momentum equation (x-direction)
where
then
121
Momentum equation (y-direction)
122
6.84: PROBLEM DEFINITION
Situation:
A torpedo-like device is being tested in a wind tunnel–additional details are pro-
vided in the problem statement.
c.s.
Drag
Force of device on air= –Drag
Find:
(a)Massrateofflow.
(b)Maximum velocity at the outlet section.
(c)Drag on the device and support vanes.
PLAN
Apply the momentum equation.
SOLUTION
Mass flow rate
At the outlet section Z0
0
vdA =Q
But vis linearly distributed, so v=vmax(r/r0).Thus
123
Momentum equation (x-direction)
c.) Outlet momentum flow
Substituting Eqns. (a) and (c) into the momentum equation (1) gives
124
Substituting numerical values into Eq. (2)
125
6.85: PROBLEM DEFINITION
Situation:
A jet engine (ramjet) takes in air, adds fuel, and then exhausts the hot gases
produced by combustion.
v1=225m/s
ρ2=0.25 kg/m3,A2=0.5m2
Find:
Thrust force produced by the ramjet: T
Assumptions:
Neglectthemassadditionduetothefuel(thatis, ˙min =˙mout =˙m=60kg/s).
2.) Assume steady flow.
PLAN
Apply the momentum equation.
SOLUTION
Force and momentum diagrams
Calculate exit velocity
Momentum equation (x-direction)
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6.86: PROBLEM DEFINITION
Situation:
Air flows through a turbofan engine. Inlet mass flow is 300 kg/s.
Bypass ratio is 2.5. Speed of bypass air is 600 m/s.
Speed of air that passes through the combustor is 1000 m/s.
m
2
m
1
.
.
300 m/s
600 m/s
1,000 m/s
c.s.
A
B
Additional details are given in the problem statement.
Find:
Thrust (T)of the turbofan engine.
Assumptions:
Neglect the mass flow rate of the incoming fuel.
PLAN
Apply the continuity and momentum equations.
SOLUTION
Continuity equation
Thus
127
Momentum equation (x-direction)
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6.87: PROBLEM DEFINITION
Situation:
Inertial reference frame.
Find:
Definition of inertial reference frame.
SOLUTION
129
6.88: PROBLEM DEFINITION
Situation:
Centrifugal acceleration on the surface of earth.
t=24h,D=8000mi.
Find:
Value of centrifugal acceleration on earth’s surface and comparison to acceleration
to gravity.
SOLUTION
The acceleration is
Acceleration
The acceleration due to gravity is 32.2 ft/s2so
130
6.89: PROBLEM DEFINITION
Maximum force occurs at the beginning; hence, the tank will accelerate immediately
after opening the cap. However, as water leaves the tank the force will decrease,
131
6.90: PROBLEM DEFINITION
Situation:
Open water tank on a frictionless plane.
Capped orifice on side has a 4-cm diameter exit pipe located 3 m below the water
surface.
Ignore all friction effects.
Find:
The force needed to keep the tank from moving when cap is removed from orifice.
PLAN
Consider the physics, and apply the momentum equation.
a. When cap is removed, a jet will result.
b. Apply Bernoulli’s eqn: find velocity of jet from head of water in tank.
c. Use momentum equation to find force needed to balance the momentum.
d. Jet and force will be x-direction only.
SOLUTION
1. Bernoulli’s equation relating 2 locations – in reservoir at depth 3m, and jet:
2. Momentum equation:
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6.91: PROBLEM DEFINITION
Situation:
A tank of water rests on a sled–additional details are provided in the problem
statement.
Find:
Acceleration of sled at time t
PLAN
Apply the momentum equation.
SOLUTION
This type of problem is directly analogous to the rocket problem except that the
weight does not directly enter as a force term and pe=patm. Therefore, the appro-
priate equation is
133
6.92: PROBLEM DEFINITION
Situation:
A cart is moving with a steady speed along a track.
Speed of cart is 5m/s(to the right). Speed of water jet is 10 m/s.
Nozzle area is A=0.002 m2.
Find:
Resistive force on cart: Fr
PLAN
Apply the momentum equation.
SOLUTION
Assume the resistive force (Fr) is caused primarily by rolling resistance (bearing
Velocity analysis
Momentum equation (x-direction)
6.93: PROBLEM DEFINITION
Situation:
Awaterjet(ρ= 1000 kg/m3)accelerates a cart
Q=0.1m3/s
Jet speed: vj=10m/s.
Cart Mass M=10kg
Deflection of the jet is normal to the cart.
Find:
(a) Develop an expression for the acceleration of the cart.
(b) Calculate the acceleration when vc=5m/s.
Assumptions:
Neglect rolling resistance.
Mass of water << mass of cart.
PLAN
To develop an equation for acceleration of the cart, apply the momentum equation to
a cv surround the cart. Select a inertial reference frame fixed to the ground because
the cart is accelerating. Then, use continuity and other equations to solve for the
acceleration.
SOLUTION
1. Force and momentum diagrams
y
mv
22
.
W
2. Momentum equation (x-direction)
5. Velocity analysis (velocity is relative to fixed reference frame)
7. Calculations
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6.94: PROBLEM DEFINITION
Situation:
A jet strikes a cart and accelerates the cart from zero to one-half the jet velocity.
Find:
Time (s) to accelerate to one-half jet velocity.
Assumptions:
No resistance to cart motion and mass of water jet moving with cart is negligible.
PLAN
Apply the momentum equation to obtain equation of motion for cart and integrate
to obtain time.
SOLUTION
Select a control surface surrounding the moving cart. Select a reference frame fixed
Momentum equation (x-direction)
dt(mvc)+ ˙m2v2x=−˙m1v1
Momentum accumulation
Note that the cart is accelerating. Thus,