A block of mass m1

is connected to an empty bucket of mass
1.35 kg by a cord over a frictionless pulley. The coefficient of
the static friction between the table and the block is 0.45 and
the coefficient of kinetic friction between the table and the
block is 0.32. Water is gradually added to the bucket until the
system just begins to move. If m1
is 28 kg;
(a) Calculate the mass of the water added to the bucket
(b) Calculate the acceleration of the entire system

To calculate the mass of water added to the bucket (part a) and the acceleration of the entire system (part b), we need to apply the principles of Newton's laws of motion.

(a) Calculate the mass of the water added to the bucket:
Let's assume the acceleration of the system is a. The net force acting on the system can be determined by considering the forces acting on the block.

The force of gravity acting on the block:
F_gravity = m1 * g

The force of static friction between the block and the table:
F_static_friction = μ_static * m1 * g (where μ_static is the coefficient of static friction)

The force of tension in the cord:
F_tension = m1 * a

Since the system is just about to move, the static friction force is at its maximum value, which is equal to the force required to overcome static friction:

F_applied = F_static_friction = μ_static * m1 * g

Solving for the mass of water added to the bucket (m_water):
m_water * g = F_applied
m_water * g = μ_static * m1 * g
m_water = μ_static * m1

Now, substitute the given values:
m_water = 0.45 * 28
m_water ≈ 12.6 kg

Therefore, the mass of water added to the bucket is approximately 12.6 kg.

(b) Calculate the acceleration of the entire system:
To find the acceleration of the system, we can consider the vertical forces acting on the bucket.

The force of gravity acting on the bucket:
F_gravity_bucket = m_bucket * g (where m_bucket is the mass of the empty bucket)

The force of tension in the cord:
F_tension = m_bucket * a

Since the bucket is not moving in the vertical direction, the net force in the vertical direction is zero:

F_net_vertical = F_tension - F_gravity_bucket = 0

Substituting the values, we get:
m_bucket * a - m_bucket * g = 0
m_bucket * (a - g) = 0

Since the mass of the empty bucket is given as 1.35 kg, we can solve for the acceleration (a):
1.35 * (a - g) = 0
a - g = 0
a = g

Given the acceleration due to gravity is approximately 9.8 m/s^2, the acceleration of the entire system is 9.8 m/s^2.

Therefore, the answers are:
(a) The mass of the water added to the bucket is approximately 12.6 kg.
(b) The acceleration of the entire system is 9.8 m/s^2.

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