Assignment 6
PHN-001 Mechanics
Submission deadline: 31st January 2023 at 12 midnight (via MS Teams only)
Q1: A 20 kg crate is at rest on a floor as shown in
the figure. The coefficient of static friction between
the crate and floor is 0.7 and the coefficient of
kinetic friction is 0.6. A horizontal force P is applied
to the crate. Find the force of friction if
a) P=20 N,
b) P= 30 N
c) P=100 N
d) P=180 N.
[Ans: a) 20N, b) 30N, c) 120N d) 118N]
Q2: A 5m ladder is resting on a smooth vertical wall with its lower end 3 m from the wall. What
should be the coefficient of friction between the ladder and the floor for equilibrium?
[Ans: 3/8]
Q3: A system consists of three identical masses m1, m2 and m3 connected by a string passing over a
pulley P. The mass m1 hangs freely and m2 and m3 are on a rough horizontal table (the coefficient of
friction=μ). The pulley is frictionless and of negligible mass. Find the downward acceleration of
mass m1.
[Ans: a=g(1-2μ)/3]
Q4:
Determine the moment of inertia of the shaded area with respect to y axis
[Ans: Iy=1333 cm4 for area1, 720 cm4 for area 2, 540 cm4 for area 3, 2593 cm4 or total area.]
Q5: A spacecraft is moving in gravity-free space along a straight path when its pilot decides to
accelerate forward. He turns on the thrusters, and burned fuel is ejected at a constant rate of 2.0×102
kg/s, at a speed (relative to the rocket) of 2.5×102 m/s. The initial mass of the spacecraft and its
unburned fuel is 2.0×104 kg, and the thrusters are on for 30 s.
a. What is the thrust (the force applied to the rocket by the ejected fuel) on the spacecraft?
b. What is the spacecraft’s acceleration as a function of time? What are the spacecraft’s
accelerations at t = 0, 15, 30, and 35 s?
[Ans: a) F= 5 ×104 N, b) a(0s)= 2.5 m/s2, a(15 s)= 2.9 m/s2, a(30 s)= 3.6 m/s2]
Q6: Obtain the moment of inertia tensor of a thin uniform ring of radius R, and mass M, with the
origin of the coordinate system placed at the center of the ring, and the ring lying in the xy plane.
Q7: The 700 N force is applied to the 100 kg block, which is
stationary before the force is applied. Determine the magnitude and
direction of the friction force F exerted by the horizontal surface on
the block.
[Ans: 379 N]
Q8: If the coefficient of friction between the steel wedge and
the moist fibers of the newly cut stump is 0.20, determine the
maximum angle α which the wedge may have and not pop out of
the wood after being driven by the sledge.
[Ans: 22.6°]
Q9: Determine the range of cylinder mass m for which the system is in equilibrium.
The coefficient of friction between the 50 kg block and the incline is 0.15 and that
between the cord and cylindrical support is 0.25.
[Ans: 12.44≦ m ≦ 78.0 kg]
Q10: Calculate the moment of inertia of the shaded area of the two overlapping
circles about the x-axis.
[Ans: 0.1988 r4]
Q11: Determine the product of inertia of the area of the quarter-circular ring
about the x-y axis. Treat the case where b is small compared with r.
[Ans: br3/2]
Q12: The double gear rolls on the stationary lower rack: the velocity of its
center A is 1.2 m/s. Determine (a) the angular velocity of the gear, and (b)
the velocities of the upper rack R and
point D of the gear.
[Ans: a) 8 s-1 b) vR=2 m/s, vD=1.697 m/s]
Q13: The 10 kg solid cylinder is resting in the inclined Vblock. If the coefficient of static friction between the
cylinder and the block is 0.50, determine (a) the friction
force F acting on the cylinder at each side before force P is
applied and (b) the value of P required to start sliding the
cylinder up the incline.
[Ans: (a) 24.5N
(b) 109.1 N]
Q14: The two blocks are placed on the incline with the
cable taut. (a) Determine the force P required to initiate
motion of the 15 kg block if P is applied down the
incline. (b) If P is applied up the incline and slowly
increased from zero, determine the value of P which
will cause motion and describe that motion.
[Ans: (a) P=56.5 N (b) When T goes slack: P=162.0
N, When T does not go slack: P=157.3 N]
Q15: The 30 kg homogeneous cylinder of 400 mm diameter rests against
the vertical and inclined surfaces as shown. If the coefficients of static and
kinetic friction between all contacting surfaces are 0.30, calculate the
applied clockwise couple M which would cause the cylinder to slip.
[Ans: NB=312 N, NA=237 N, M=32.9 N-m]
Q16: A floor joist (50 mm x 200 mm) has a 25-mm hole drilled
through it to for a water pipe installation. Determine the
percentage reduction n in moment of inertia of the cross-sectional
area with respect to the x-axis (compared with that of the
undrilled joist). Then, evaluate this expression for y = 50 mm.
[Ans: n = 0.1953 + 0.00375y2 , n=9.57%]
Q17: The rectangular area shown in part a of the figure is
split into three equal areas which are then arranged as shown
in part b of the figure. Determine an expression for the
moment of inertia of the area in part b about the centroidal xaxis. What percent increase n over the moment of inertia for
area a does this represent if h = 200 mm and b = 60 mm ?
[Ans: Ix = 110.4 (106) mm4 , n = 176.0%]
Q18: An 80.0 kg gymnast dismounts from a high bar. He
starts the dismount at full extension, then tucks to
complete a number of revolutions before landing. His
moment of inertia when fully extended can be
approximated as a rod of length 1.8 m and when in the
tuck a rod of half that length. If his rotation rate at full
extension is 1.0 rev/s and he enters the tuck when his
center of mass is at 3.0 m height moving horizontally to
the floor, how many revolutions can he execute if he
comes out of the tuck at 1.8 m height?
[Ans: 4.0 rev/s]
Q19: The friction tongs shown are used to lift a 350 kg casting. Knowing that h = 864 mm,
determine the smallest allowable value of the coefficient of static friction between the casting and
blocks D and D’.
[Ans: 0.19]
Q20: A woman pedals her bicycle up a 5-percent grade and a slippery road at a steady speed. The
woman and bicycle have a combined mass of 82 kg. with mass centre at G. If the rear wheel is on
the verge of slipping, determine the coefficient of friction μs between the rear tire and the road. If
the coefficient of friction is doubled, what would be the friction force acting on the rear wheel?
(why may we neglect friction under the front wheel)
[Ans: μs =0.082]
Q21: The coefficients of friction are μs = 0.4 and μk = 0.3 between all surfaces of contact.
Determine the force P for which motion of the 30-kg block is impending if cable AB (a) is attached
as shown, (b) is removed. Assume that the cord is inextensible and the pulleys are well oiled.
[Ans: a) 361 N, b) 196.2 N]
Q22: Two persons A and B, each of mass m are standing at the two ends of rail-road car of mass
M. The person A jumps to the left with a horizontal speed u with respect to the car. Thereafter, the
person B jumps to the right, again with the same horizontal speed u with respect to the car. Find the
velocity of the car after both the persons have jumped off.
[Ans: m2u/((M+2m)(M+m))]
Q23: A uniform rod AB of mass m and length 5a is free to rotate on a smooth horizontal table
about a pivot through P, a point on AB such that AP = a. A particle of mass 2m moving on the table
strikes AB perpendicularly at the point 2a from P with speed v, the rod being at rest. If the
coefficient of restitution between them is
!
"
, find their speeds immediately after impact.
[Ans: ω=15v/(37a)]
Q24: A uniform disc having radius 2R and mass density σ as shown in figure. If a small disc of
radius R is cut from the disc as shown. Then find out the moment of inertia of remaining disc
around the axis that passes through O and is perpendicular to the plane of the page.
[Ans: 13 σπR4/2]
Q25: A uniform cylinder of radius 0.5m and weight 90 kg rolls
without slipping down a 45°-incline and drags the 45 kg block B with
it. What is the kinetic energy of the system if block B is moving at a
speed of 5m/sec? Neglect the mass of connecting agents between the
bodies.
[Ans: 347.975 (SI)]
Q26: Determine the moment of inertia for the shaded area about the x and yaxis. Given a = 2 m and b = 4 m.
[Ans: IX = 39.0 unit; IY = 8.53 unit]
Q27: A Platform rotates at an angular speed of w1, while a cylinder of radius
‘r’ and length ‘a’ mounted on the platform at an angular speed of w2. When
the axis of the cylinder is collinear with the stationary Y axis, what is the
angular momentum vector of the cylinder about the centre of mass of the
cylinder? The mass of the cylinder is M.
!
$% ! &' !
[Ans: # 𝑀𝑟 # 𝜔# 𝑗 + 𝜔! '
!#
' #
+ (𝑐 + #* + 𝑘 ]
Q28: Determine the moment of inertia Iu ,Iv and the product of inertia
Iuv for the rectangular area. The u and v axes pass through the centroid
C.
[Ans: Iu = 1.28(106) mm4; Iv = 3.31(106) mm4;
Iuv = -1.75(106) mm4]
Q29: A slab A weighting 2 kN rides on two rollers each having a mass
of 50kg and a radius of gyration of 200 mm. If the system starts from
rest, what minimum constant force T is needed to prevent the slab from
exceeding the speed of 3m/s down the incline in 4sec? There is no
slipping.
[Ans: 1.061 kN]
Q30: The polar moment of inertia for the area is Jc=642 !"106 mm4, about the z
axis passing through the centroid C. The moment of inertia about the y axis is
264"! 106 mm4, and the moment of inertia about the x axis is 938"! 106 mm4.
Determine the area A.
[Ans: 14 x 103 mm2]
Q31: Part of the conveyor system is shown. A link belt is meshed around
portions of gears A and B. The belt has a mass of 5 kg/m. and a radius of 300mm.
If a force of 100 N is applied at one end as shown, What is the maximum possible
¨ = 5 𝑟𝑎𝑑⁄𝑠𝑒𝑐 # ? The gears
force T that can be transmitted at the other end if 𝜃
turn freely.
[Ans: 28.34 N]
Q32: A reference xyz is moving such that the origin O has at time t a
velocity relative to reference XYZ given as
𝑉( = 6𝑖 + 12𝑗 +
13𝑘 𝑓𝑡⁄𝑠𝑒𝑐
The xyz reference has an angular velocity 𝜔 relative to XYZ at
time t given as
𝜔 = 10𝑖 + 12𝑗 + 2𝑘 𝑟𝑎𝑑 ⁄𝑠𝑒𝑐
a) What is the time rate of change relative to XYZ of a
directed line segment 𝜌 going from position (3,2,-5) to (-2,
4, 6) in xyz?
What is the time rate of change relative to XYZ of position vectors 𝑖 ) ∧ 𝑘 ) ?
[Ans: (a) 128i – 120j + 80k; (b) -12k + 2j; 12i – 10j]
Q33: A missile travels in a straight line with respect to inertial reference
XYZ at speed V1=10ft/sec and a speed a1=5ft/sec2. At the same instant the
missile rolls about its direction of flight at an angular speed 𝜔! =
5 𝑟𝑎𝑑⁄𝑠𝑒𝑐 and change in angular speed 𝜔˙! = 5 𝑟𝑎𝑑 ⁄𝑠𝑒𝑐 # . Inside the
missile a rod is rotating at a constant angular speed (relative to the missile)
𝜔# = 10 𝑟𝑎𝑑 ⁄𝑠𝑒𝑐 about an axis perpendicular to the page. Find the
velocity and acceleration of the tip of the rod relative to XYZ at an instant
shown.
[Ans: VXYZ = 20i – 30j ft/sec; aXYZ = 20i + 5j – 500k]
Q34: Determine the moment of inertia of the homogeneous triangular prism with
respect to the y axis. Express the result in terms of the mass m of the prism. Hint:
For integration, Use thin plate elements parallel to the x-y plane having a thickness
of dz.
[Ans:
*
+
(𝑎# + ℎ# ) ]
Q35: The area of the cross section of an airplane wing has the following properties
about the x and y axes passing through the centroid C: 𝐼, = 450 𝑖𝑛" , 𝐼- =
1730 𝑖𝑛" , 𝐼,- = 138 𝑖𝑛" . Determine the orientation of the principal axes and the
principal moments of inertial.
[Ans: Imax = 1.74"! 103 in4; Imin = 435 in2]
Q36: Estimate the acceleration of a rocket at the moment when the spacecraft reaches the orbit.
Take the following parameters: the specific impulse (exhaust velocity) is u = 3000 vm/s, the mass of
the spacecraft on the orbit is m = 5000 kg, the fuel burn rate is 𝜇 = 100 kg/s.
[Ans: 60 m / s2]