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.
To calculate the mass of sand added to the bucket, we need to determine the point where the system just begins to move. At this point, the force of friction on the block is equal to the force of static friction. We'll use the following steps to solve for the mass of sand added:
Step 1: Identify the forces acting on the block:
- Weight of the block (mg)
- Tension in the cord (T)
- Force of static friction on the block (fs)
Step 2: Write the equations for each force:
- Tension in the cord (T) is equal to the weight of the block (mg) plus the weight of the bucket (mbg):
T = (m + mb) * g
- The force of static friction on the block (fs) is given by:
fs = μs * N
where μs is the coefficient of static friction, and N is the normal force acting on the block. The normal force is equal to the weight of the block (mg):
N = mg
Step 3: Set up the equation for motion:
We'll use Newton's second law, which states that the net force is equal to the mass of the system (m + mb) multiplied by the acceleration (a):
T - fs = (m + mb) * a
Step 4: Substitute the equations from Step 2 into the equation from Step 3:
(m + mb) * g - μs * mg = (m + mb) * a
Step 5: Simplify the equation:
First, distribute the g to both terms on the left side:
mg + mbg - μs * mg = (m + mb) * a
Then, isolate the variables on one side:
mbg - μs * mg = (m + mb) * a - mg
Finally, divide by a to solve for mb:
mb = [(m + mb) * a - mg] / (g - μs * g)
Now, to calculate the acceleration of the system, we can use the equation for motion that we derived in Step 3:
T - fs = (m + mb) * a
Substituting the known values:
[(m + mb) * g] - (μk * mg) = (m + mb) * a
Since the system is just beginning to move, the force of kinetic friction is equal to the force of static friction, so we can use the coefficient of static friction (μs) instead of the coefficient of kinetic friction (μk).
Simplifying the equation:
(m + mb) * g - (μs * mg) = (m + mb) * a
Now, we can solve these equations simultaneously to find the mass of sand added to the bucket and the acceleration of the system.