A 28.0 kg block is connected to an empty 2.10 kg bucket by a cord running over a frictionless pulley. The coefficient of static friction between the table and the block is 0.44 and the coefficient of kinetic friction between the table and the block is 0.27. Sand is gradually added to the bucket until the system just begins to move.
Calculate the acceleration of the system
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To calculate the acceleration of the system, we need to compare the forces acting on the block and the bucket.
First, let's consider the block. The forces acting on the block are the force of gravity (mg) and the force of static friction (fs) or kinetic friction (fk) if the block is already in motion.
Since we want to find the acceleration when the system just begins to move, the block is on the verge of sliding and static friction is at its maximum. Therefore, the force of static friction (fs) is given by:
fs = coefficient of static friction * normal force
The normal force (N) is the force exerted by the table on the block and is equal to the weight of the block, which is given by:
N = mg
Substituting this into the equation for the force of static friction:
fs = coefficient of static friction * mg
Now, let's consider the bucket. The forces acting on the bucket are the force of gravity (mg) and the tension in the cord (T).
Since the bucket is empty, there is no sand pulling it down, so the force of gravity is simply mg.
Now, assuming the system is moving upwards (as sand is added to the bucket), we can set up the following equations of motion:
For the block:
ma = fs - mg
For the bucket:
Ma = T - mg
Since the block and the bucket are connected by the same cord, they have the same acceleration (a = A). Therefore, we can rewrite the second equation as:
ma = T - mg
Now, let's substitute the expression for the force of static friction (fs) into the equation for the block:
ma = (coefficient of static friction * mg) - mg
Simplifying this equation, we get:
ma = (coefficient of static friction - 1) * mg
Lastly, since we know the masses of the block (m = 28.0 kg) and the bucket (M = 2.10 kg), we can substitute these values into the equation to find the acceleration (a):
a = [(coefficient of static friction - 1) * mg] / (m + M)
Now, you can calculate the acceleration by plugging in the given values for the coefficient of static friction, m (mass of the block), M (mass of the bucket), and g (acceleration due to gravity).