A 28.0 block is connected to an empty 2.20 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 0.35. 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.
Well, well, well, it seems we have a system that's just begging to be puzzled over. Don't worry, I'm here to bring some laughter into the mix.
Let's figure this out step by step. First, we need to determine the maximum force of static friction. We can do this by multiplying the coefficient of static friction (0.44) by the normal force (the weight of the block, which is mass multiplied by gravity).
Once we have the maximum force of static friction, we can compare it to the force applied by the bucket of sand. If the force applied by the sand is greater, we're dealing with the coefficient of kinetic friction (0.35) and dynamic movement. But if the force applied by the sand is equal to the maximum force of static friction, then it's just enough to get the block moving.
To calculate the mass of sand added to the bucket, we can use Newton's second law, which states that force equals mass times acceleration. In this case, the force applied by the sand is the maximum force of static friction. So, we can divide that force by the acceleration to get the mass of the sand.
Once the sand is added, the system will accelerate. To find the acceleration, we need to take into account the force applied by the sand and subtract the force of kinetic friction (which is the coefficient of kinetic friction multiplied by the normal force).
So, let's put on our math caps and calculate these values step by step. But remember, even if physics makes you want to scream, just laugh it off!
Good luck!
To calculate the mass of the sand added to the bucket, we need to determine the limiting friction force between the block and the table.
1. Calculate the weight of the block:
Weight = mass * acceleration due to gravity
= 28.0 kg * 9.8 m/s^2
≈ 274.4 N
2. Calculate the maximum static friction force:
Static friction force = coefficient of static friction * Normal force
= 0.44 * Weight
≈ 0.44 * 274.4 N
≈ 120.54 N
3. Calculate the mass of sand added to the bucket:
Mass of sand = Force of sand / acceleration due to gravity
= (Static friction force - Weight) / acceleration due to gravity
= (120.54 N - 274.4 N) / 9.8 m/s^2
≈ -15.88 kg
Note: The negative sign indicates that adding this amount of mass of sand will cause the system to start moving.
To calculate the acceleration of the system:
4. Calculate the kinetic friction force:
Kinetic friction force = coefficient of kinetic friction * Normal force
= 0.35 * Weight
≈ 0.35 * 274.4 N
≈ 95.94 N
5. Calculate the net force acting on the system:
Net force = Applied force - Friction force
= Weight of sand - Kinetic friction force
= (Mass of sand + Mass of bucket) * acceleration due to gravity - Kinetic friction force
= (-15.88 kg + 2.20 kg) * 9.8 m/s^2 - 95.94 N
≈ -196.41 N
6. Calculate the acceleration of the system:
Acceleration = Net force / (Mass of block + Mass of bucket)
= (-196.41 N) / (28.0 kg + 2.20 kg)
≈ -6.728 m/s^2
Note: The negative acceleration indicates that the system will move in the opposite direction due to the sand being added to the bucket.
To solve this problem, we need to consider the forces acting on the system.
1. Calculate the mass of sand added to the bucket:
Since the system is at the point of just beginning to move, the force of static friction between the block and the table exactly balances the force applied by the hanging block due to gravity. This can be represented by the equation:
Static friction force = Force due to hanging block
The force due to the hanging block is equal to the weight of the block, which is given by:
Force due to hanging block = mass of the hanging block * acceleration due to gravity
Now, the static friction force can be calculated using the equation:
Static friction force = coefficient of static friction * normal force
The normal force can be determined by considering the forces acting on the hanging block:
Normal force = Force due to hanging block
Substituting these equations, we get:
coefficient of static friction * normal force = mass of block * acceleration due to gravity
Now, we can solve for the mass of the sand added to the bucket:
mass of sand = (coefficient of static friction * mass of block * acceleration due to gravity) / acceleration due to gravity
2. Calculate the acceleration of the system:
Once the system starts moving, the static friction changes to kinetic friction. The force of kinetic friction can be calculated using the equation:
Force of kinetic friction = coefficient of kinetic friction * normal force
The normal force will be the same as we calculated earlier. Since the force of kinetic friction opposes the motion of the system, it is equal to the force applied by the hanging block. Therefore, we have:
Force of kinetic friction = Force due to hanging block
Now, we can solve for the acceleration:
Force of kinetic friction = mass of sand + mass of block * acceleration
Rearranging the equation, we get:
acceleration = (Force due to hanging block) / (mass of sand + mass of block)
Substituting the value for the force due to the hanging block, we can calculate the acceleration.
Remember to use the correct values for the coefficient of static friction, the coefficient of kinetic friction, the mass of the block, and the acceleration due to gravity to get accurate results.
.0125
Wb = mg = 28.0kg * 9.8N/kg = 274.4N. =
Weight of block.
W2 = mg = 2.20kg * 9.8N/kg = 21.56N. =
Weight of bucket.
Fb = 274.4N @ 0deg.,
Fp = 274.4sin(0) = 0 = Force parallel to table.
Fv = 274.4cos(0) = 274.4N. = Force
perpendicular to table.
Ff = u*Fv = 0.44 * 274.4 = 120.7N =
Force of static friction.
Ff' = 0.35 * 274.4 = 96.04N. = Force of kinetic friction.
1. Fn=(Fap+F2) - Fp - Ff = 0 = Net force,
(Fap+21.56) - 0 - 120.7 = 0,
Fap = 120.7 - 21.56 = 99.14N. = Force
applied = Force added to bucket.
Fap = mg = 99.14N.,
9.8m = 99.14,
m = 10.1kg = mass added to the bucket.
2. Fn = Fap + F2 - Ff',
Fn = 99.14 + 21.56 - 96.04 = 24.66N.
a = (Fp-Fap) / (m1+m2),
a = (0-99.14 / (28+12.3) = -2.46m/s.