Sheet 7 Kinetics of Particles
Newton’s second law and equations of motion for the particle
Summary:
Linear Momentum:
𝑃
The linear momentum P can be expressed as:
𝑃 = 𝑚𝑣
Newton’s Second Law of Motion:
𝐹 = 𝑚𝑎
Where: F is the resultant of the forces acting on the particle.
a is the acceleration of the particle.
m is the mass of the particle.
The equation of the Newton’s second law is a vector equation and its components
along the two components of the accelerations can be written as:
∑ 𝐹𝑥 = 𝑚𝑎𝑥
&
Friction Force:
It is one of the forces acting on the
particle during its motion on a rough
surface. It is parallel to the rough
surface and opposite to the motion.
∑ 𝐹𝑦 = 𝑚𝑎𝑦
From the Lab experiments:
 F<Fmax the particle is in the static
case.
 F=Fmax=μsN the particle is about to
move (the last stage of static).
 F=μkN the particle is moving.
 μs = static coefficient of friction.
 μk = kinetic coefficient of friction.
Note:
The Free body diagram (F.B.D.) must be drawn before solving the problem
explaining all the forces acting on the body, direction of acceleration and direction
of axes.
Problems:
F13.2 If motor M exerts a force F = (10 t2 +100) N on
the cable, where t is in seconds, determine the velocity
of the 25-kg crate when t = 4 sec. The coefficient of
static and kinetic friction between the crate and the
plane are μs = 0.3 and μk = 0.25, respectively. The crate
is initially at rest.
F13.2 A spring of stiffness k = 500 N/m is
mounted against the 10-kg block. If the block
is subjected to the force of F = 500 N,
determine the velocity at s = 0.5 m. When s =
0, the block is at rest and the spring is
uncompressed.
The contact surface is
smooth.
General Problems:
1) The block of mass 5 kg slides
down on an inclined plane of
angle 𝜃 and moves with constant
velocity when 𝜃 = 30o. Find the
acceleration of the block when 𝜃
= 45o.
2) A box of 80 kg slides on a rough
horizontal surface by a force P.
When P increases the block starts
to slide. Find the initial
acceleration that the box will
have if μs=0.5 and μK=0.3.
3) A block of mass 60 kg is places
on a rough horizontal surface
of μs=0.5 and μK=0.25. If a
force F is applied to the block
and the block starts to move at
t = 4 sec., find the constant B.
Find the velocity of the block
when t = 8 sec.