Download Exercise of Mechanics Set 4 (A) --- Kinetics of a Particle ---

Exercise of Mechanics
Set 4 (A) Kinetics of a Particle
1.
Prove the following formulas by considering that integration is a reverse process of
differentiation.
d cos x
 - sin x,   sin x dx   cos x
dx
d tan x
 sec 2 x   sec 2 x dx  tan x  C
dx
d cot x
 - csc 2 x   csc 2 x dx   cot x  C
dx
dx n
x n1
 n x n-1
  x n dx 
 C, n  - 1
dx
n 1
d ln x 1
1

  dx  ln x  C
dx
x
x
x
de
 ex
  e x dx  e x  C
dx
C is a constant
2.
A log B of mass 200 kg is to be pulled up a 30o ramp (Fig. 1). The block A of mass 125
kg is released from rest in the position shown. The coefficient of kinetic friction
between the log and the ramp is 0.5. Neglecting the weights of the pulley C and the
ropes, determine the velocity of the log when A hits the ground 6 m below.
Ans.: 2.31ms-1
C
A
B
30o
3.
Fig. 1
6m
The chain of total length L is released from rest with an overhanging length b (Fig. 3).
Neglecting friction, determine the velocity of the chain when its last link leaves the edge
of the table.
Ans.: v = [gl (1- b2/l2)]1/2
b
4.
Fig. 2
A car of mass 1500 kg enters a section AC of a curved horizontal road and slows down
at a uniform rate from a speed of 100 km h-1 at A to a speed of 50 km h-1 at C. The
distance traveled by the car between A and C is 200 m. The radius of curvature of the
road at A is 400 m and at C is 80 m, and the centers of curvature at the two points are on
opposite sides of the road. Point B, between A and C, is the inflection point where the
1
curvature changes direction. Find the friction exerted by the road on the wheels at point
A, B and C.
Ans.: 3.62 kN; 2.17 kN; 4.22 kN
5.
A large horizontal disc of a proposed amusement ride is designed to rotate about a
vertical axis through its centre with a constant angular speed of 0.5 rad s-1. Assuming
that a man of mass 68 kg is walking outwards along a radius marked on the disc at a
constant speed of 0.91 m s-1, calculate the radial and transverse horizontal forces that the
disc exertion on the man when he is 3.0 m from the center of the disc.
Ans.: Fr = -51 N; F= 62 N
6.
A smooth bar AB rotates in a vertical plane at a constant angular velocity of 1.5 rad s-1 in
the anticlockwise direction (Fig. 3). Collar C of mass 0.4 kg is sliding inward relative to
the bar at a speed of 3 m s-1 when it is 0.8 m from the hinge. Find (a) the force exerted
on C by bar AB, and (b) the acceleration of C relative to AB at the instant shown.
Ans.: (a) 1.64 N (b) 6.70 ms-2
B
0.8 m
C
Fig. 3
o
60
A
1.5 rad s-1
7.
An object is given an initial velocity of 7.75 ms-1 up the incline at point A (Fig. 4). The
coefficient of friction between the object and the incline is 0.3. Calculate the ratio of the
normal reaction exerted on the object by the supporting surface just after the object
passes point B to that before it reaches B.
Ans. : 1 : 2
B
7.75 m s
A
8.
9.
-1
2m

2m
Fig. 4
4m
A satellite is moving in an elliptical orbit around the earth. When it reaches a point at a
height of 500 km above the surface of the earth, its velocity is found to be 3.00x104 km
h-1. Calculate its velocity when it reaches another point 1200 km above the surface of the
earth. (Radius of the earth Re = 6.371 x 106 m)
Ans.: 7.66 x 103 m s-1

A particle of mass 1.2 kg moves in a vertical plane under the influence of a force F and


gravitational field. F varies with time t. The momentum of the particle is given by G =
9 (3 + t2) x̂ + 4 (4- t3) ŷ , where all quantities are measured in SI units, and the unit
vectors x̂ and ŷ are pointing in the horizontal and vertically upwards directions

respectively. Determine the vector F as a function of time and also its magnitude
when t = 2 s.

Ans.: F = 18t x̂ + (11.8 - 12 t2) ŷ ; 51.1 N
2