Question

A plank of length l = 2m is hinged at one end to a wall. The other end is being (temporarily) supported by a worker who is holding it up with his hand, keeping the plank horizontal. The plank has a mass of 20kg, and there is also a toolbox of mass 5kg sitting on it, 50cm away from the worker (1.5 m away from the wall). (a) Draw a free body diagram and an extended free-body diagram for the plank. (b) What are the magnitudes of (1) the upwards force exerted by the worker on the plank and (2) the force at the hinge?(c) If the worker were to let go of the plank, what would its angular acceleration be as it starts swinging down? The moment of inertia is I = 1Ml2. (Note: assume the toolbox stops pressing down on the plank immediately. This is a good approximation, as you shall see below.)(d) Consider a point on the plank located immediately below the toolbox. As the plank swings, this point moves in a circle of radius 1.5 m. What is its linear (tangential) acceleration as it starts going down, and how does it compare to the acceleration of gravity?

Answer 1

Given,

Mass of the plank, M = 20 kg

Mass of the toolbox, m = 5 kg

Length of the plank, L = 2 m

a)

Let F be the force exerted by the worker

Let f be force at the hinge

b)

There, is no motion, hence,

= 0

=> 0 = F + f - Mg - mg

=> F + f = Mg + mg

Now,

About point A,

= 0

=> 0 = Mg*1 + mg*1.5 - F*2

=> F*2 = Mg + 1.5*m*g

=> F*2 = 20g + 1.5*5*g = 27.5 g

=> F = 27.5/2 *g = 13.75 *g = 13.75 * 9.8

         = 134.75 N

Force exerted by the worker is 134.75 N

Now,

F + f = Mg + mg

=> 13.75g + f = 20g + 5g = 25g

=> f = 25g - 13.75g = 11.25g

=> f = 11.25*9.8 = 110.25 N

or force at hinge is 110.25 N

c)

Assume that the toolbox stops the pressing the plank as soon as plank is let go.

Let angular acceleration be

Now,

= I

=> I = Mg*1 = 20*g

Since,

I = 1/3*M*L² = 1/3*20*2*2 = 80/3 kg m²

Thus,

=> 80/3* = 20*g

=> = 3/4g = 0.75*9.8 = 7.35 rad/s²

or angular acceleration is 7.35 rad/s²

d)

We have,

= 7.35 rad/s²

distance of the point, R = 1.5 m

Now,

Tangential acceleration, a = R = 7.35 * 1.5

                                           = 11.025 m/s²

Tangential acceleration is more than the acceleration of gravity.