Transcript CONNECTED PARTICLES - Gayaza High School

CONNECTED PARTICLES
LESSON THREE
By
Ronald Ddungu
[email protected]
MOTION ON A ROUGH INCLINED
PLANE
A particle of mass 4kg rests on the surface of a
rough plane which is inclined at 300 to the
horizontal. It is connected by a light inelastic
string passing over a light smooth pulley at the
top of the plane, to a particle of mass 6kg which
is hanging freely. If the coefficient of friction
between the 4kg mass and the plane is 0.45, find
the acceleration of the system when it is released
from rest and the tension in the string. Find also
the force exerted by the string on the pulley.
DIAGRAM
SOLUTION(All forces acting are shown)
a ms-2
a ms-2
Fr
Equations of motion
Consider the 4kg mass;
There is no motion
perpendicular to the
plane. Thus resolving
perpendicular to the
plane gives;
R – 4gcos300 = 0
R = 4g x √3/2
R= 2g√3N……(i)
There is an acceleration
ams-2 up the plane.
The resultant force parallel to
the plane is;
T – 4gsin300 –Fr .
Thus T – 4gsin300 –Fr = 4a..(ii)
But Fr = μR
= 0.45(2g√3)N
= 9g√3N ……………(iii)
10
Calculations
Using (iii) in (ii)
T – 4gsin300 –9g√3 = 4a
10
T – 2g – 9g√3 = 4a…….(iv)
10
For the 6kg mass assumed
to have a resultant force
vertically downwards;
6g –T = 6a…………(v)
Adding (iv) and (v)
T – 2g – 9g√3 = 4a
10
+ 6g –T = 6a
g(4 - 0.9x √3) = 10a
10a = 2.4411x 9.8
a = 2.3923ms-2
Finding the tension
6g –T = 6a
T = 6g – 6a
T = 6x(9.8 - 2.3923)
T = 44.4462N
Therefore the acceleration
of the system is
2.3923ms-2 and the
tension in the string is
44.4462N
Force exerted on the pulley is
through the tension as shown
below
300
Finding the resultant force
Resolving the forces;
• Resultant force
• Horizontally( );
R
66.6693N
-Tcos300 = - 44.4462cos300
= - 38.49263N
38.49263N
• Vertically( );
R2 = 38.492632 + 66.66932
-Tsin300 - T = - 1.5T
R2 = 5926.478127
R = √(5926.478127)
= -1.5 X44.4462 R = 76.9836N
Inclined at an angle
= - 66.6693N
Tan-1 (66.6693/38.49263) = 60.00 .
The force exerted by the string onto the
pulley is 76.9836N inclined at 600 to the
horizontal.
HOME WORK
DIAGRAM
QUESTION
A particle of mass M kg rests on the surface of a
rough plane which is inclined at 300 to the
horizontal. It is connected by a light inelastic
string passing over a light smooth pulley at the
top of the plane, to a particle of mass N kg which
is hanging freely. If the coefficient of friction
between the M kg mass and the plane is 0.35,
find the acceleration of the system when it is
released from rest and the tension in the string.
Find also the force exerted by the string on the
pulley in the cases below.
(a) When M = 5 and N = 7
(b) When M = 8 and N = 5
RESEARCH
• http://www.youtube.com/watch?v=jHVRF5c6I4 (Watch this video)
• http://www.mei.chosenhill.org/Solutions/mec
hanics1/Revision%20Connected%20Particles.p
df (Down load this document and bring it with
you to school)
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