A 26.0 kg block is connected to an empty 1.00 kg bucket by a cord running over a frictionless pulley. The coefficient of static friction between the table and the block is 0.490 and the coefficient of kinetic friction between the table and the block is 0.320. Sand is gradually added to the bucket until the system just begins to move.
Part A of the question asked me to calculate the mass of sand added to the bucket which i determined to be 11.7 kg.
Part B asks the acceleration of the system downward. Can someone help?
Wb = mg = 26kg * 9.8N/kg = 254.8N.
Fb = 254.8N @ 0deg.,
Fp = 254.8sin(0) = 0 = Force parallel to table.
Fv = 254.8cos(0) = 254.8N. = Force perpendicular to table.
Fs = u*Fv = 0.490 * 254.8 = 124.9N. =
Force of static friction.
Fk = 0.320 * 254.8 = 81.5N. = Force of
kinetic friction.
A. Fap - Fp - Fs = 0,
Fap - 0 - 124.9 = 0,
Fap = 124.9N. = Force applied = Force of sand plus bucket.
mg = Fap,
m = Fap/g = 124.9/9.8 = 12.7kg = mass of bucket plus sand.
Ms = 12.7 - 1 = 11.7kg = Mass of sand.
B. Fn = Fap - Fp - Fk,
Fn = 124.9 - 0 - 81.5 = 43.4N. = Net
force.
a = Fn / (m1+m2) = 43.4 / (26+12.7) =
1.12m/s^2.
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To find the acceleration of the system downward, we can use Newton's second law of motion: Fnet = ma.
The net force acting on the system is the difference between the gravitational force pulling the block downward and the frictional force opposing its motion.
The gravitational force pulling the block downward can be calculated by multiplying the mass of the block by the acceleration due to gravity (9.8 m/s²).
F_gravity = (26.0 kg)(9.8 m/s²) = 254.8 N
The frictional force opposing the block's motion when it's at the verge of moving is equal to the product of the coefficient of static friction and the normal force.
F_friction = (coefficient of static friction)(normal force)
The normal force is equal to the weight of the bucket and the added sand combined:
Normal force = (mass of bucket + mass of sand)(acceleration due to gravity)
Normal force = (1.00 kg + 11.7 kg)(9.8 m/s²) = 126.98 N
F_friction = (0.490)(126.98 N) = 62.2752 N
Now, the net force can be calculated:
Fnet = F_gravity - F_friction
Fnet = 254.8 N - 62.2752 N = 192.5248 N
Finally, we can find the acceleration using Newton's second law:
Fnet = ma
192.5248 N = (26.0 kg + 1.00 kg + 11.7 kg)(a)
192.5248 N = 38.7 kg(a)
a = 4.9804 m/s²
So, the acceleration of the system downward is approximately 4.9804 m/s².
To calculate the acceleration of the system downward, we can use Newton's second law of motion. The net force acting on the system is equal to the mass of the system multiplied by the acceleration.
The net force can be determined by considering the forces acting on the block and the bucket separately.
For the block:
- The force of gravity acting downward is given by the mass of the block multiplied by the acceleration due to gravity (9.8 m/s^2).
- The static friction force is given by the coefficient of static friction multiplied by the normal force acting on the block. The normal force is equal in magnitude and opposite in direction to the force of gravity acting on the block.
Thus, the static friction force on the block is given by the coefficient of static friction multiplied by the force of gravity.
For the bucket:
- The force of gravity acting downward is given by the mass of the bucket multiplied by the acceleration due to gravity.
- The tension in the cord is acting upward and it is equal in magnitude to the force of gravity acting on the block.
Since the system is just beginning to move, the static friction force on the block is equal to the force of gravity acting on the block.
Let's calculate the net force:
For the block:
Force of gravity = (26.0 kg) * (9.8 m/s^2) = 254.8 N
Static friction force = (0.490) * (254.8 N) = 124.9552 N
For the bucket:
Force of gravity = (1.00 kg) * (9.8 m/s^2) = 9.8 N
Since the tension in the cord is equal to the force of gravity acting on the block, the tension is also 254.8 N.
The net force can be calculated by summing the forces acting on the system:
Net force = Tension - Static friction force - Force of gravity (bucket)
Net force = 254.8 N - 124.9552 N - 9.8 N = 120.0448 N
Finally, we can calculate the acceleration using Newton's second law:
Net force = (Total mass of the system) * (acceleration)
Total mass of the system = mass of the block + mass of the bucket + mass of sand added
Total mass of the system = 26.0 kg + 1.00 kg + 11.7 kg = 38.7 kg
acceleration = Net force / Total mass of the system
acceleration = 120.0448 N / 38.7 kg ≈ 3.10 m/s^2
Therefore, the acceleration of the system downward is approximately 3.10 m/s^2.
To calculate the acceleration of the system downward, you can start by finding the net force acting on the system.
The net force can be determined by subtracting the force due to friction from the force due to gravity.
1. Force due to gravity: The force due to gravity is given by the equation: F_gravity = mass * acceleration_due_to_gravity.
For the 26.0 kg block:
F_gravity_block = 26.0 kg * 9.8 m/s^2.
For the 1.00 kg bucket:
F_gravity_bucket = 1.00 kg * 9.8 m/s^2.
2. Force due to friction: The force due to friction depends on whether the system is in motion or at rest.
a) If the system is at rest:
The force of static friction opposes the impending motion (just before it starts moving), and its magnitude is given by the following equation: F_static_friction = coefficient_of_static_friction * normal_force.
The normal force can be calculated by summing the forces perpendicular to the table. Since both the block and the bucket are on the table surface, the normal force is equal to the force due to gravity.
For the 26.0 kg block:
F_normal_block = F_gravity_block = 26.0 kg * 9.8 m/s^2.
For the 1.00 kg bucket:
F_normal_bucket = F_gravity_bucket = 1.00 kg * 9.8 m/s^2.
Therefore, the force due to static friction is:
F_static_friction = coefficient_of_static_friction * (F_normal_block + F_normal_bucket).
b) If the system is in motion:
Once the system starts moving, the force of friction switches to kinetic friction. The magnitude of kinetic friction is given by the equation: F_kinetic_friction = coefficient_of_kinetic_friction * normal_force.
Similar to the static friction case, the normal force is equal to the force due to gravity.
Therefore, the force due to kinetic friction is:
F_kinetic_friction = coefficient_of_kinetic_friction * (F_normal_block + F_normal_bucket).
3. Net force:
To find the net force, subtract the force due to friction from the force due to gravity:
Net_force = (F_gravity_block + F_gravity_bucket) - (F_static_friction or F_kinetic_friction).
Finally, apply 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:
Net_force = (mass_block + mass_bucket) * acceleration.
Setting the net force equations equal to each other, we can solve for acceleration:
(mass_block + mass_bucket) * acceleration = (F_gravity_block + F_gravity_bucket) - (F_static_friction or F_kinetic_friction).
Substituting the known values, you can solve for acceleration.