To determine the largest mass that can hang without forcing the block to slide, we need to consider the forces acting on the block.
The first force we need to consider is the force of gravity. The weight of the block is given by the formula W = m * g, where m is the mass of the block and g is the acceleration due to gravity.
W = 4.00 kg * 9.8 m/s^2
W = 39.2 N
Next, we need to consider the force of static friction. The formula for static friction is given by the formula Ff = µ * N, where µ is the coefficient of static friction and N is the normal force.
In this case, the normal force is equal to the weight of the block, since the block is at rest on the horizontal table.
N = W
N = 39.2 N
Now we can calculate the maximum force of static friction:
Ff = µ * N
Ff = 0.54 * 39.2 N
Ff = 21.168 N
The force of static friction acts in the opposite direction of the force trying to make the block slide. If the hanging mass is too large, the force due to the hanging mass will be greater than the force of static friction, and the block will start to slide.
Since the force due to the hanging mass is transmitted through the string and the pulley to the block, the maximum force due to the hanging mass that can be supported without forcing the block to slide is equal to the force of static friction.
Therefore, the largest mass that can hang in this way without forcing the block to slide is equal to the force of static friction divided by the acceleration due to gravity:
m = Ff / g
m = 21.168 N / 9.8 m/s^2
m ≈ 2.16 kg
So, the largest mass that can hang in this way without forcing the block to slide is approximately 2.16 kg.