Problem 5.24
A particle is traveling in a straight line in a horizontal plane. The mass of the particle is 0.10 kg.
The acceleration of the particle is described by the following equation:
a=
dV
= A − Bt 2
dt
where A = constant and B = 6 m/s4. The following additional information is known about the
motion of the particle:
At t = 0:
x = x0 = 8 m, V= V0 = 0
and
At t = 1 second:
V = 30 m/s
(a) Determine the time(s) at which the velocity is zero.
(b) Determine the total distance traveled when t = 5 sec. (Note that the change in position x is
not the same as the distance traveled.)
(c) Plot the acceleration, velocity, and position of the particle for 0 ≤ t ≤ 5 seconds . Determine
the time(s) when the motion of the particle changes direction.
(d) For the same time interval as in Part (c), plot the linear momentum of the particle, the rate of
change of linear momentum of the particle, and the net external force on the particle in the
horizontal plane. Determine the time(s) when the net external force changes direction.
(e) Discuss how the time(s) you calculated in Part (c) when the particle changed its direction of
motion relate to the time(s) you calculated in Part (d) when the net external force changes
direction. Is there any relation?
Problem 5.25 (Adapted from Engineering Mechanics: Statics by Bedford & Fowler, AddisonWesley)
25 m
25 m
13 m
B
C
A
A 140-kg traffic light is suspended above the street by two cables as shown in the figure above.
On a windy day the wind blows from left to right on the figure and creates a horizontal force of
200 N on the traffic light. Assume that the deflection of the traffic light from a vertical
orientation due to the crosswind is negligible.
(a) Starting with the rate-form of the conservation of linear momentum, use the closed system
shown by the dashed lines in the figure to solve for the tension in the cables AB and AC with
and without the crosswind force.
(b) How would your analysis change for (a) if your system only included the intersection of the
two cables and the traffic light as shown inside the light gray dashed circle? How would you
handle the force applied by the hanging traffic light?
Problem 5.26
A 30-kg package is placed on an incline when a force P is applied to it as shown on the figure.
The coefficients of static and kinetic friction between the package and the incline are 0.2 and 0.1,
respectively. The motion of the package depends on the magnitude of the force P.
(a) Determine the range of values for P for
which the package will remain stationary on
the inclined plane.
(b) Assuming the package is initially stationary,
determine its velocity and position 10
seconds after the force P is reduced to zero,
i.e. P = 0.
P
φ = 50o
θ = 20o
Problem 5.27
The mass of Block A is 30 kg and the mass of Sphere B
is 5 kg. Block A slides on the surface and as it slides
Sphere B is free to move as shown in the figure. The
coefficient of kinetic friction between Block A and the
surface is 0.24. Assume that the link connecting the
Block and the Sphere has negligible mass.
(a) Determine the numerical value of the force F, if the
angle θ = 20o is constant.
(b) Is the linear momentum of the block increasing,
decreasing, or constant? Is the block accelerating,
decelerating, or moving at constant velocity?
Block
F
Sphere
θ
(c) Determine the magnitude and direction of the force of the link acting on the Sphere. Is there
tension or compression in the link connecting Block A and Sphere B?
Problem 5.28
In a cathode-ray tube, an electron with mass m enters the gap between two charged plates at Point O with
a velocity V = Vo i. While it is between the charged plates, the electric field generated by the plates
subjects the electron to a force F = - e E j where e is the charge of the electron and E is the electric field
strength. The plate spacing is 2h, the length of the charged plates is l, and the distance to the screen is L.
(See figure below.) Assume that gravitational forces can be neglected and that external forces on the
electron are negligible when it is not between the charge plates.
(a) Starting with the appropriate conservation and accounting relations, develop an expression for the
location of the point of impact of the electron with the screen. Your answer should be presented in
terms of the initial speed of the electron (Vo) , the mass of the electron (m), the charge of the electron
(e), the electric field strength (E) , and the dimensions h , l , and L . Clearly identify your system and
show how you use the material in the problem and any additional assumptions to develop your
answer.
(b) Sketch the path of the electron on the figure below.
(c) Calculate the actual deflection if the following numerical values are known:
Vo = 2.2 × 107 m/s;
L = 100 mm;
m = 9.11 × 10-31 kg;
e = 1.6 × 10-19 C (coulombs);
E = 15 kN/C
l = 30 mm;
y
Charged Plates
Screen
- - - - 2h
O
x
+ + + +
l
L
Problem 5.29
A block with a mass m = 200 kg rests on an inclined plane with
an applied load P. Depending upon the magnitude of P, the
block may move up the incline, down the incline, or remain
stationary. Between the surfaces of contact, the coefficient of
static friction is µs = 0.25 and the coefficient of kinetic friction
is µk = 0.20.
Determine the range of the magnitude of the horizontal force P,
in newtons, that will keep the block in equilibrium, i.e. not
moving.
θ = 60o
P
Problem 5.30
The ducted fan unit shown in the figure has mass m
= 100 kg and is supported in the vertical position on
its flange at A. The unit draws in air with a density
ρ = 1.200 kg/m3 and a velocity V1 = 5 m/s through
an inlet with diameter D1 = 1.00 m. It discharges air
through two outlets at the bottom of the fan. The
mass flow rate through each outlet is 1/2 of the
entering mass flow rate, and the velocity at each
outlet is V2 = V3 = 15 m/s. Both inlet and outlet
pressures are atmospheric.
Determine the vertical force R, in newtons, applied
to the flange of the fan unit by the supports.
D1 = 1.00 m
V1 = 5 m/s
A
Fan Blades
V2 = 15 m/s
V3 = 15 m/s
θ = 55o
θ = 55o
Problem 5.31
A crate with mass m = 500 lbm is attached to a motorized winch and positioned on a sloping dock with
inclination angle θ = 30o as shown in the figure. The winch exerts a tensions force T on the crate through
the winch cable. The friction coefficients between the crate and the dock surface are µS = 0.30 for static
friction and µK = 0.25 for kinetic friction.
Initially, the crate is stationary and the winch brake is on to prevent any motion. Suddenly at t = 0, the
brake is released and the winch exerts a constant tension force T = 100 lbf on the crate. At t > 2 seconds,
the tension force exerted by the winch on the crate suddenly increases to a constant value T = 400 lbf
(a) Consider the forces on the stationary crate when the
winch brake is on and find the value or range of
values for the tension force T under these conditions?
(b) Consider the crate’s motion for 0 ≤ t ≤ 2 s when the
tension force is T = 100 lbf. Find the crate’s
acceleration, velocity and position at t = 2 seconds.
(c) Consider the crate’s motion for the period t > 2 s
when the tension force is T = 400 lbf. Qualitatively
describe how the crate moves during this period (Be
concise; use words not numbers!)
(d) Calculate the impulse for the tension force T over the
time interval t = 0 to 4 seconds. Be sure to indicate
both the direction and magnitude.
Winch Operating Conditions
t<0
: Winch locked and crate stationary
0≤t≤2s
: Winch operating; T = 100 lbf.
t > 2 s : Winch operating; T = 400 lbf
where T is the tension force of the winch on the crate.
Winch cable
Winch
m = 500 lbm
θ = 30o
µS = 0.30; µK = 0.25
Problem 5.32
You have been asked to investigate the performance of a jet-propelled boat using a water channel where
the water velocity Vwater can be varied as required. The boat is placed in the channel and tethered so that it
is stationary. The boat is jet-propelled by a pump that develops a constant volumetric flow rate of water,
Vpump . Water enters the aft (front) of the boat through an area of A1 and leaves at the stern (rear) through
an area A2.
Water flowing over the hull of the boat exerts a drag force on the boat in the direction the water is
flowing. This horizontal drag force which includes the net pressure forces on the hull is given by the
following equation:
2
Fdrag = kVwater
where k is a constant.
Assume that the angle θ and water density ρ are both known.
a) Find expressions for the water velocities V1 and V2 in terms of the pump flow rate, Vpump .
b) Find an expression for the volumetric flow rate through the pump Vpump as a function of the water
= f (V ) .
velocity in the channel, Vwater, when the tension in the tether is zero, i.e. V
pump
water
Tether
Pump
θ
(2)
(1)
Vwater
Problem 5.33
A 10-kg steel sphere is suspended from a 15-kg
frame, and the sphere-frame combination slides
down a 20o incline as shown in the figure. The
coefficient of kinetic friction between the frame
and the incline is µk = 0.15.
o
A 45
45o
B
Determine the tension in each of the supporting
wires, in newtons.
20o
Problem 5.34
You have been hired by NASCAR to analyze vehicle
impacts into the wall. For the crash at right, solve for
the average reactions (Rx,avg and Ry,avg) of the wall on the
car in terms of the mass of the vehicle m, the initial
velocity V1, the angle θ, the distances h and d, and the
time interval ∆ t = t2-t1 assuming the vehicle comes to a
complete stop during the time interval.
Top View
h
y
x
d
Feel free to assume that the car remains a rectangle
during the impact.
V1
θ
Problem 5.35
Blocks A and B are identical and each have a mass of 10 kg. Block B is at rest when it is hit by block A
which is moving with velocity VA = 6 m/s just before impact. Blocks A and B stick together after the
impact, and you may neglect friction during the impact.
After the impact, the velocity of blocks A and B decreases due to friction. The coefficient of kinetic
friction between all surfaces is µk = 0.20.
(a) Determine the velocity of block A and B immediately after A hits B.
(b) Determine the impulse of the force of block A on block B during the impact.
(c) Determine the time required for the velocity of the blocks to drop to 1 m/s.
(d) Determine the distance traveled by the blocks during this time interval.
Before the impact
After the impact, blocks A and
B travel together.
VA = 6 m/s
A
B
A
B
Problem 5.36
A piece of wood is held at rest on the smooth (frictionless) inclined plane by the stop block at A. A bullet
is traveling as shown in the figure with velocity V when it becomes embedded in the block. The
embedding takes a short time, ∆t . The mass of the wood is mw, the mass of the bullet is m B, the angle of
the inclined plane is θ and the acceleration due to gravity is g.
Provide symbolic solutions to answer the following questions:
(a) What is the velocity Va of the bullet/wood
immediately after the bullet becomes embedded?
SET UP BUT DO NOT SOLVE. Clearly show how
you would use your equations to solve for Va
For the remaining questions, you may assume that Va is
known.
(b) What is the average impulsive force acting on the
bullet during the impact?
(c) What is the equation for the rate of change of
velocity of the bullet/wood after the impact?
Problem 5.37
Salt water (ρ = 1025 kg/m3) enters a vertical pipe at a
volumetric flow rate of 0.5 m3/s and is discharged into
the atmosphere from the two 30o outlets as shown in the
figure. The flow divides equally between the two outlets.
Each of the discharge nozzles has an outlet diameter of
10.0 cm and the inside diameter of the pipe at section AA is 25 cm. The pressure of the water at section A-A is
550 kPa and the atmospheric pressure is 100 kPa. The
pipe above the flange and the water within it has a mass
of 60 kg.
Determine the total force exerted by the lower pipe on
the section of pipe above the flange A-A. Indicate both its
magnitude, in newtons, and its direction.
30O
30O
A
A
Salt water flows
into the pipe.
Problem 5.38
A jet engine with exhaust nozzle is mounted on a test stand as shown in the figure. The engine is mounted
as shown on two hangars and a diagonal brace. All connections are with frictionless pinned joints.
Information about the flow area, pressure, and air velocity at three locations along the engine are given in
the table. At steady-state operation, air is sucked into the inlet at the rate of 30 kg/s.
Determine the direction and the magnitude of the force T in the diagonal brace. Is the brace in tension or
compression?
1m
Sec. 1
2
Sec. 2
Sec. 3
Flow area
m
0.15
0.16
0.06
Pressure
kPa
84
240
114
Air velocity
m/s
120
315
600
1m
60o
60o
T
1
2
Problem 5.39
A bullet strikes and glances off a flat plate as shown in the figure. The plate is resting on a frictionless
horizontal surface. Initially, the bullet has a velocity of 800 ft/s and the plate is stationary. After striking
the plate, the bullet velocity is 600 ft/s.
Determine the final velocity of the plate.
mBullet = 0.05 lbm
mPlate = 3 lbm
30o
15
o
3
Problem 5.40
A truck is traveling down a long steady grade (θ = 15o) as shown in the figure. The crate has mass mCrate
= 500 kg with height H = 1 m and length L = 2 m. The crate rests on the trailer bed and the surface has a
static friction coefficient µ S = 0.4 and a kinetic friction coefficient µ K = 0.3. To prevent the load from
shifting, the driver must limit his braking.
Determine the maximum truck deceleration if the crate does not slide on the trailer bed.
mCrate = 500 kg
L=2m
H=1m
θ = 15o
Problem 5.41
Two blocks rest on an inclined plane with
θ = 35o as shown in the figure. Block A
has mass mA = 13.5 kg and block B has
mass mB = 40 kg. The coefficients of
static and kinetic friction between all
surfaces are µS = 0.3 and µk = 0.2,
respectively. Initially the blocks are
stationary and are supported by a stop
block and fixed-length wire, as shown on
the figure.
Wire
When the stop block is removed the
block B immediately starts to move
because the angle θ is large enough to
produce motion.
Find the acceleration of block B and the
tension in the wire immediately after the
stop block is removed.
A
B
Stop
Block
θ
Problem 5.42
The convertible shown is moving at a constant
velocity, vc, in the direction shown. At the instant
shown a passenger in the car throws a ball from the car.
The magnitude and direction of the initial velocity of
the ball with respect to the car, v0, is shown in the
figure. The ball hits the ground 30 ft to the right of the
point from which it was released.
10 ft/s
45°
4.5 ft
vc
a) How long does it take the ball to hit the ground from the moment it is released?
b) What is the velocity of the car?
Problem 5.43
The cart has mass M and is filled with water that has a
mass m0. If a pump ejects water through a nozzle having a
cross-sectional area A at a constant rate of v0 relative to the
cart, determine the velocity of the cart as a function of
time. What is the maximum speed developed by the cart
assuming all the water can be pumped out? Assume the
frictional resistance to forward motion is F and the density
of water is ρ.
Problem 5.44
Two swimmers A and B, of mass 75 kg and 50 kg,
respectively, dive off the end of a 200-kg boat. Each swimmer
has a relative horizontal velocity of 3 m/s when leaving the
boat. If the boat is initially at rest, determine its final velocity,
assuming that (a) the two swimmers dive simultaneously, (b)
swimmer A dives first, (c) swimmer B dives first.
B
A