A 4.00 kg block is pushed along the ceiling with a constant applied force of 85.0 N that acts at an angle of 55.0° with the horizontal, as in the figure below. The block accelerates to the right at 6.00 m/s2. Determine the coefficient of kinetic friction between block and ceiling.
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To determine the coefficient of kinetic friction between the block and the ceiling, we can follow these steps:
Step 1: Analyze the forces acting on the block.
- Start by drawing a free-body diagram to identify the forces.
- In this case, the only horizontal force acting on the block is the applied force (F_applied) of 85.0 N.
- The weight of the block (mg) acts vertically downwards.
- The normal force (N) acts perpendicular to the ceiling.
- The kinetic friction force (f_k) acts opposite to the direction of motion.
Step 2: Resolve the applied force into components.
- Since the applied force is at an angle with the horizontal, we need to determine its horizontal and vertical components.
- The horizontal component (F_horizontal) is given by F_applied * cos(theta), where theta is the angle of 55.0°.
- The vertical component (F_vertical) is given by F_applied * sin(theta).
Step 3: Find the normal force.
- The normal force (N) is equal to the weight of the block, which is given by N = mg.
- Since the block is on the ceiling, the normal force is acting upward.
Step 4: Calculate the frictional force.
- The frictional force (f_k) is determined by the equation f_k = mu_k * N, where mu_k is the coefficient of kinetic friction.
- Since the block is accelerating, the frictional force is equal to the mass of the block multiplied by its acceleration (f_k = m * a).
Step 5: Equate the horizontal forces.
- The sum of the horizontal forces must be equal to the mass of the block multiplied by its acceleration (F_horizontal - f_k = m * a).
- Plug in the values and solve for F_horizontal.
Step 6: Solve for the coefficient of kinetic friction.
- Once you have the value for F_horizontal, plug it into the equation from Step 5 to calculate the coefficient of kinetic friction (mu_k = (F_horizontal - m * a) / N).
Using these steps, you can determine the coefficient of kinetic friction between the block and the ceiling.