A 30.0-kg block is resting on a flat horizontal table. On top of this block is resting a 15.0-kg block, to which a horizontal spring is attached, as the drawing illustrates. The spring constant of the spring is 340 N/m. The coefficient of kinetic friction between the lower block and the table is 0.640, and the coefficient of static friction between the two blocks is 0.870. A horizontal force vector F is applied to the lower block as shown. This force is increasing in such a way as to keep the blocks moving at a constant speed. At the point where the upper block begins to slip on the lower block determine the following.
To determine the following, we will need to calculate the force required to keep the blocks moving at a constant speed, and then compare it to the maximum static friction force between the blocks.
1. Calculate the force required to keep the blocks moving at a constant speed:
The force required can be determined using Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration.
The acceleration of the system is zero since the blocks are moving at a constant speed. Therefore, the net force acting on the system is zero.
Since there are no vertical forces acting on the blocks, we will only consider the horizontal forces.
The forces acting horizontally on the system are:
- The applied force vector F.
- The force of kinetic friction acting on the lower block.
- The force of static friction acting between the two blocks.
The force of kinetic friction can be calculated using the formula:
Force of kinetic friction = coefficient of kinetic friction * normal force.
The normal force exerted on the lower block is equal to its weight, which can be calculated as the mass of the lower block multiplied by the acceleration due to gravity (9.8 m/s^2).
The force of static friction can be calculated using the formula:
Force of static friction = coefficient of static friction * normal force.
When the upper block is just starting to slip, the force of static friction is at its maximum.
2. Compare the force required to the maximum static friction force:
If the force required to keep the blocks moving at a constant speed is greater than the maximum static friction force, then the upper block will slip. If the force required is less than or equal to the maximum static friction force, then the blocks will not slip.
To determine the maximum static friction force, multiply the coefficient of static friction by the normal force exerted on the upper block.
The normal force exerted on the upper block can be calculated as the sum of the weight of the upper block and the weight of the lower block.
Now, you can calculate the values for the force required to keep the blocks moving and the maximum static friction force and compare them to determine if the upper block will slip or not.