CONNECTED PARTICLES

LESSON THREEBy

Ronald Ddunguronaldddungu@yahoo.com

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 10T – 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 = 6aT = 6g – 6aT = 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;• Horizontally( );-Tcos300 = - 44.4462cos300

= - 38.49263N• Vertically( ); -Tsin300 - T = - 1.5T = -1.5 X44.4462 = - 66.6693N

• Resultant force

66.6693N

38.49263N

R2 = 38.492632 + 66.66932

R2 = 5926.478127R = √(5926.478127)R = 76.9836NInclined at an angle 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.

R

HOME WORKDIAGRAM

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=jHVRF-5c6I4 (Watch this video)

• http://www.mei.chosenhill.org/Solutions/mechanics1/Revision%20Connected%20Particles.pdf (Down load this document and bring it with you to school)

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