Calculate the acceleration of the system.?

Question

A 31.0 kg block is connected to an empty 1.00 kg bucket by a cord running over a frictionless pulley (Fig. 4-57). The coefficient of static friction between the table and the block is 0.425 and the coefficient of kinetic friction between the table and the block is 0.320. Sand is gradually added to the bucket until the system just begins to move.

Answer 1

First solve for the force of friction between the block and the table using the formula,

f=uR, where f is the force of friction, u the coefficient of static friction, and R is the force normal to the table. R therefore is the weight of the block equal to 31g.

f=0.425*31*9.8
=129.1N

The weight of the sand added to the bucket plus the weight of the bucket itself should be just slightly equal to 129.1N so that the system will begin to move. At that point, however, the block will begin to accelerate because the force of friction acting on it will no longer be static friction but kinetic friction, which is lower in magnitude. Thus there will be a net force equal to 129.1 minus the force of kinetic friction on the block. Thus,

Net force, F=129.1-0.320*31*9.8
=129.1-97.2
=31.9N

F=ma, where F is our net force of 31.9N, m the total mass of block plus mass of sand and bucket, and a is the acceleration of the system.

Mass of sand and bucket=129.1/9.8
=13.2kg
Substitute known values:

31.9=(31+13.2)a
a=31.9/44.2
=0.72m/s^2

Answer 2

If the block is just beginning to move, then the force of the bucket and sand must have just matched the friction force of the block on the table, using the coefficient of static friction. This force is 31kg * 9.8m/s^2 * .425 = 129.115N. The amount of mass in the bucket & sand object is 129.115 / 9.8 = 31 * .425 = 13.175kg.

Now the coefficient of kinetic friction, .320, takes over with the block, so that friction force of the block is now less than it was. So we now have a net force in the direction of the bucket, and the system will accelerate towards the bucket. To get the acceleration of the system, you would have to calculate the net force on the system, then divide by the total mass of the system. The positive force of this is due to the weight of the bucket & sand, and the negative force is due to the block and the coefficient of kinetic friction.