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M2 Rigid Bodies

(Q6 Edexcel M2 June '06)

A wooden plank AB has mass 4m and length 4a. The end A has the plank lies on a rough horizontal ground. A small stone of mass m is attached to the plank at B. The plank is resting on a small smooth horizontal peg C, where BC = a. The Plank is in equilibrium making an angle 'alpha' with the horizontal, where tanalpha = 3/4. The coefficient of friction between the plank and the ground is u. The plank is modelled as a uniform rod lying in a vertical plane perpendicular to the peg, and the stone as a particle.

Show that;

a. the reaction at the peg on the plank has magnitude 16/5 mg

b. u >= 48/61

I got the first bit right, but I'm having trouble with (b).

mev.JPG14.5KB

Reply 1

RodT

Let R = normal reaction at ground, S = Normal reaction at peg, then:

Vertically, forces balance so R + S cos(alpha) = 5mg

Horizontally, friction is overcoming any motive force (because plank is not moving)so uR >= S sin(alpha)

Moments about A balance, so
2a (4mg cos(alpha)) + 4a (mg cos(alpha)) = 3a S

As tan(alpha)=3/4, cos(alpha)=4/5 and sin(alpha)=3/5.

The moment-balancing equation gives S = 16/5 mg

The two force balance equations then give u >= 48/61.

Hope this helps... :smile: