CONNECTED PARTICLES
LESSON FOUR
By
Ronald Ddungu
[email protected]
Movable Pulley System
A
B
M
NOTES
• In the pulley system shown above, pulley A is
fixed and pulley B is free to move. If the free
mass M is pulled down then pulley B can
move up. For the pulley B to move up a
distance of x m then a string of 2x m must
pass over the fixed pulley A and allow the free
mass to go down 2xm.
Analysis of the motion
• If we assume the free
mass and the movable
pulley to be released
from rest and then be
allowed to travel for a
time of 1scovering
distances 2x m and x m
respectively;
Then using S=ut + 1/2at2 .
For the pulley;
x = 0(1) + 1/2a(12 )
a = 2x …………………….(i)
• For the free mass;
• 2x =0(1) + 1/2a(12 )
• a = 4x …………………….(ii)
• Thus the acceleration of
the free mass is always
twice the acceleration of
the movable pulley. That
is to say if the free mass
moves at 2ams-2 then the
movable pulley will move
at ams-2
EXAMPLE
• A string with one end fixed , passes under a
moveable pulley of mass 3kg ,over a fixed
pulley and carries a 6kg mass at its other end
as shown in the diagram below. Find the
acceleration of the moveable pulley and the
6kg mass. Also find the tension in the string.
DIAGRAM
a ms-2
T
3kg
T
2a ms-2
T
3g
6kg
6g
Equations of motion
• Let the acceleration of the For the 6kg mass;
3kg pulley be ams-2
6g – T = 6(2a)……….(i)
upwards then the 6kg free For the 3kg pulley;
mass will move downwards
2T – 3g = 3a………….(ii)
at 2ams-2
Solving these simultaneously;
• 2x (i) + (ii)
12g -2T = 24a
+ 2T - 3g = 3a
9g = 27a
a = 9g/27
a = g/3 ms-2
2a = 2g/3ms-2
Finding the tension
From (ii)
The 6kg mass goes down with
-2 = 6.53333ms-2
acceleration
2g/3ms
2T – 3g = 3a
and the 3kg pulley goes up
2T = 3a + 3g
-2 = 3.26667ms-2.
with
g/3
ms
2T = 3(a + g)
The
tension
in
the
string
is
19.6N.
2T = 3(g/3 +g)
2T = 3(4g/3)
T =2g
T = 2 x 9.8
T = 19.6 N
HOME WORK
A string with one end fixed , passes under a
moveable pulley of mass P kg ,over a fixed
pulley and carries a Q kg mass at its other end
as shown in the diagram below. Find the
acceleration of the moveable pulley and the Q
kg mass. Also find the tension in the string in
each of the cases below.
(i) P = 4 and Q = 7
(ii) P = 1 and Q = 3
DIAGRAM
A
P kg
Q kg
RESEARCH
• https://sites.google.com/site/appliedmathsrat
oath/connected-ipartles/multiple-pulleysystems (Watch the video on moveable pulley
problem at the bottom of the page that loads)
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