Q1. (Marks 40/200)
A carriage of mass M is at rest on a smooth rigid floor as
shown in figure Q1. A point mass m is attached to a slender rigid of length L
with negligible mass is pivoted to the carriage at point A trough a
frictionless pin. The point mass & rod release from the rest in horizontal
position at time t =0. At time t = T seconds the rod is at vertical position as
shown in dotted lines in figure Q1. M = 5 mB and mB=10 kg and L=1.5 m.
i. Determine the absolute velocity of the carriage and the
point mass B when the rod is at vertical position.
(15 Marks) ii. Determine the distance travel by carriage
during time T seconds. (15 Marks)
iii. What is the angular velocity of the mass B at time T
seconds (10 Marks)
Figure Q1
Q2. (Marks 40/200)
At the instant shown in the Figure Q2, link AB has an
angular velocity ?AB = 3 rad/s. angular acceleration aAB = 2 rad/s2. Each link
is considered as a uniform slender bar each with a mass of 2.0 kg/m,
I. Determine the angular velocity and angular acceleration
of link BC and CD (20 Marks)
II. Determine the kinetic energy of each link (AB, BC and
CD) (10 Marks)
III. Determine the horizontal and vertical components of
acceleration at point C (10 Marks)
Figure Q2
Q3. (Marks 40/200)
Figure Q3 shows a 10Mg front-end-loader used to move Ore in
the loader bucket. The centres of mass (CM) for the front-end-loader and the
Ore load at G and GL respectively.
I. Determine the reactions exerted by the ground on the
pairs of wheels at A and B, if the front-end-loader is moving forward in a
inclined surface (170 to horizontal) at a constant acceleration of 0.5 m/s2
from the rest. The Ore Load is 2.5Mg and the CM (GL) is at height h = 3.5 m and
x = 2.0 m (from the centre of front wheel A). (20 Marks)
II. What is the maximum acceleration can develop by the
front end loader to move forward on the inclined plane without tipping over.
(10 Marks)
III. Investigate the motion of the front end loader case (i)
if the inclination of the surface become 260 to the horizontal and the surface
condition with loose soil i.e static friction between the wheels and the ground
is 0.43. (10 Marks)
Figure Q3
Q4. (Marks 40/200)
A slender uniform bar AB shown in Figure Q4 has a mass of 5
kg and length L = 2.5m and at rest in vertical plane. A thin 1.5 kg circular
disk of radius 100 mm is fixed to the bar AB through a smooth pin at it centre
B. Two springs support the rod at A, and C, AC = 0.8L. The rod is pivoted to
ground using a frictionless pin at point E and it oscillates about its
equilibrium position. K1=K2= 1600 N/m. A dashpots which have a damping
coefficient 600 N s/m is attached to the rod AB at D, AD=0.35L as shown in
Figure Q4.
(i) Write the equation of motion for small oscillations of
the rod& disk attachment about its equilibrium position (20 Marks)
(ii) Determine the damp-free natural frequency of vibration
of the bar & attached disk arrangement. (10 Marks)
(iii) Calculate damping coefficient ? (damping factor) and
comment on the damping condition. (10 Marks)
Elevation
Figure Q4
Q5. (Marks 40/200)
Figure Q5 shows a collision of a high speed locomotive
engine and an empty oil tanker at an unprotected railway crossing. The
locomotive A has mass MA and was travelling in constant velocity of 100 km/hr
and the Tanker B has mass MB and was travelling in constant velocity of 80
km/hr. MA = 4.5 Mg and MB = 1.3 Mg
I. Calculate velocities of the locomotive and the tanker
after the collision if the locomotive and the tanker become entangled and move
off together after the collision (no-derailment happened). (10 Marks)
II. Calculate velocities of the locomotive and the tanker
after the collision in terms of e (0 e 1) which the coefficient of restitution
between the locomotive and the tanker. (10 Marks)
III. Calculate the possible energy loss for Case (i) and
Case (II) for values of e = 0.8. Comment on the severity of collision depending
on your calculated values. (10 Marks)
IV. In the Case ii with e=0.8 , estimate the angular
velocity of the tanker if the tanker has started rotating (approx) about point
O just after it moving away from the locomotive. Take the radius of gyration k
of the empty tanker (about point O) = 2.48 m (10 Marks)
List all the assumptions clearly for each case.
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