A 26 kg block is connected to an empty 1 kg bucket by a cord running over a frictionless pulley. The coefficient of static friction between the table and the block is 0.435 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.
Now, I'm supposed to find the mass of sand added to the bucket and the acceleration. What's throwing me off is the static friction and what equations to use. HELP?
To solve this problem, let's break it down step by step:
1. Identify the objects and forces involved:
- The 26 kg block (object A)
- The 1 kg bucket (object B)
- The cord connecting them
- The force of gravity on both objects
- The normal force on object A
- The force of static friction between object A and the table
- The force of kinetic friction between object A and the table
2. Determine the conditions for the system to start moving:
- The force of static friction must be overcome for the system to begin moving.
- The maximum force of static friction can be calculated using the coefficient of static friction.
3. Calculate the maximum force of static friction:
- Maximum force of static friction, F_static_max = coefficient of static friction × normal force
- Normal force, N = mass A × g (where g is the acceleration due to gravity)
4. Compute the force due to gravity on object A:
- Force due to gravity on object A, F_gravity_A = mass A × g
5. Determine the force required to overcome static friction:
- Force required to overcome static friction, F_required = F_gravity_A
6. Find the minimum force applied to move the system:
- Minimum force applied to move the system = F_required = F_gravity_A
7. Determine the force on object B:
- Force on object B due to the weight of the sand, F_sand = mass_sand × g
8. Calculate the acceleration of the system:
- Acceleration, a = (F_sand - F_required) / (mass A + mass B)
9. Use Newton's second law to relate acceleration, force, and mass:
- F_net = (mass A + mass B) × a
10. Finally, determine the mass of sand added to the bucket:
- Mass of sand added to the bucket = F_sand / g
By following these steps and using the provided coefficients of friction and masses, you can find the mass of sand added to the bucket and the acceleration of the system.