You are lowering two boxes, one on top of the other, down a ramp by pulling on a rope parallel to the surface of the ramp. Both blocks move with constant velocity of 15 m/s. The coefficient of kinetic friction between the ramp and the lower box is 0.444 and the coefficient of static friction between the two boxes is 0.800. What is the magnitude and direction of the frictional force on the upper box? Magnitude of the applied force? What angle will the top block start sliding down the bottom block?
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To find the magnitude and direction of the frictional force on the upper box, as well as the magnitude of the applied force, we need to analyze the forces acting on the system.
Let's break it down step by step:
1. Start by drawing a free-body diagram of the upper box. The only horizontal force acting on it is the frictional force, which opposes the motion. Therefore, the magnitude of the frictional force on the upper box can be found using the formula:
frictional force = coefficient of kinetic friction * normal force
Since the box is moving with constant velocity, the normal force equals the weight of the box (mg), where m is the mass of the box and g is the acceleration due to gravity.
2. To find the normal force for the upper box, we need to consider the forces acting on the lower box. The normal force acting on the lower box can be found by subtracting the weight of the upper box from the weight of the lower box:
normal force = (mass of lower box * gravity) - (mass of upper box * gravity)
3. Now that we have the normal force for the upper box, we can calculate the magnitude of the frictional force by multiplying it with the coefficient of kinetic friction provided.
4. To find the direction of the frictional force, we can use the concept of opposite directionality. Since the boxes are moving with constant velocity and the frictional force always opposes the motion, the frictional force on the upper box will be in the opposite direction of the applied force.
5. The magnitude of the applied force can be determined by balancing the forces acting on the system. Since the boxes are moving with constant velocity, the applied force will be equal to the frictional force on the lower box. Therefore, we can use the same formula as before:
frictional force = coefficient of static friction * normal force
Since the boxes are not accelerating, the normal force here will be the total weight of both boxes (mg).
6. To find the angle at which the top block will start sliding down the bottom block, we need to consider the forces acting on the top block. The force of static friction between the blocks provides the necessary force to maintain the top block at rest. When this force is overcome, the top block will start sliding. Therefore, we can set up the equation:
force of static friction = maximum static friction force = coefficient of static friction * normal force
Since the normal force here is the weight of the top block (mg), we can solve for the angle by rearranging the equation and isolating the angle.
By following these steps and plugging in the given values, you should be able to find the answers to all three parts of the question.