A 28.0kg block is connected to an empty 1.80kg bucket by a cord running over a frictionless pulley. The coefficient of static friction between the table and the block is 0.41 and the coefficient of kinetic friction between the table and the block is 0.35. Sand is gradually added to the bucket until the system just begins to move .
Calculate the mass of sand added to the bucket.
Calculate the acceleration of the system.
Start with a free-body diagram:
http://img297.imageshack.us/img297/3702/1254156394brittany.jpg
The tension in the cord is the sum of the weight of the bucket (m2) and the sand (m3), i.e.
T=(m2+m3)g
To start motion, T >= μs*N ....(1)
where
μs = the coeff. of static friction
N = normal reaction from the table
=m1*g
m1=mass of the block
Using equation (1), m3 can be isolated and calculated (<10 kg).
Check my thinking and calculations.
To calculate the mass of sand added to the bucket, we need to determine the minimum force required to overcome static friction and set the block and bucket in motion. The force required to overcome static friction is given by the equation:
Fs = μs * N
where Fs is the force of static friction, μs is the coefficient of static friction, and N is the normal force acting on the block.
The normal force N is equal to the weight of the block, which is given by:
N = m * g
where m is the mass of the block and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting the values into the equations:
N = 28.0 kg * 9.8 m/s^2 = 274.4 N
Fs = 0.41 * 274.4 N = 112.504 N
So, the force required to overcome static friction is 112.504 N.
Since the same force is also acting on the bucket, the mass of the sand added to the bucket can be calculated using the equation:
m = Fs / g
where m is the mass of the added sand.
Substituting the values:
m = 112.504 N / 9.8 m/s^2 ≈ 11.48 kg
Therefore, the mass of sand added to the bucket is approximately 11.48 kg.
To calculate the acceleration of the system, we can use Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration:
ΣF = m * a
In this case, the net force is the difference between the force of static friction and the force of kinetic friction, which is given by:
ΣF = Fs - Fk
where Fs is the force of static friction and Fk is the force of kinetic friction.
The force of kinetic friction is given by:
Fk = μk * N
where μk is the coefficient of kinetic friction and N is the normal force.
Substituting the values:
N = 1.80 kg * 9.8 m/s^2 = 17.64 N
Fk = 0.35 * 17.64 N = 6.174 N
ΣF = 112.504 N - 6.174 N = 106.33 N
Since the net force is equal to the mass of the system multiplied by its acceleration, we can rearrange the equation to solve for acceleration:
a = ΣF / (m1 + m2)
where m1 is the mass of the block and m2 is the mass of the added sand.
Substituting the values:
a = 106.33 N / (28.0 kg + 11.48 kg) ≈ 2.42 m/s^2
Therefore, the acceleration of the system is approximately 2.42 m/s^2.