A block of mass 3.42 kg is drawn at constant speed a distance of 1.32 m along a horizontal floor by a rope exerting a constant force of magnitude 8.62 N and making an angle of 17.0 degrees above the horizontal. Compute the coefficient of friction between the block and the floor.
horizontal force=friction=8.62cos17
vertical force on block up=8.62sin17
friction force=mu(3.42g-8.62sin17)
set this equal to the horizontal force, and solve for mu
To compute the coefficient of friction between the block and the floor, we can use the equation:
frictional force = coefficient of friction * normal force
First, let's calculate the normal force on the block. The normal force equals the weight of the block since it is on a horizontal floor and not accelerating vertically:
normal force = weight = mass * gravitational acceleration
The gravitational acceleration on Earth is approximately 9.8 m/s^2. Plugging in the values:
normal force = 3.42 kg * 9.8 m/s^2 = 33.516 N
Next, let's calculate the horizontal component of the tension force exerted by the rope. This component counteracts the frictional force:
horizontal component of tension = tension force * cos(angle)
Plugging in the values:
horizontal component of tension = 8.62 N * cos(17.0 degrees) = 8.62 N * 0.9537 = 8.218 N
Since the block is moving at a constant speed, the frictional force equals the horizontal component of the tension force:
frictional force = 8.218 N
Finally, we can use the equation for the frictional force:
frictional force = coefficient of friction * normal force
Plugging in the values:
8.218 N = coefficient of friction * 33.516 N
Solving for the coefficient of friction:
coefficient of friction = 8.218 N / 33.516 N ≈ 0.245
To compute the coefficient of friction between the block and the floor, we need to use Newton's second law of motion and analyze the forces acting on the block.
Step 1: Determine the Normal Force
The normal force is the force exerted by the floor on the block perpendicular to the surface. Since the block is on a horizontal floor and is not accelerating vertically, the normal force is equal in magnitude and opposite in direction to the force of gravity. Therefore, the normal force can be calculated as:
Normal Force = mass * gravity
Where:
- mass = 3.42 kg (given)
- gravity = 9.8 m/s^2 (acceleration due to gravity)
Step 2: Determine the Frictional Force
The frictional force can be calculated using the equation:
Frictional Force = coefficient of friction * Normal Force
Where:
- coefficient of friction = unknown
- Normal Force = calculated in step 1
Step 3: Determine the Horizontal Force
The horizontal force can be determined by decomposing the applied force into its horizontal and vertical components. The horizontal component can be calculated as:
Horizontal Force = applied force * cos(angle)
Where:
- applied force = 8.62 N (given)
- angle = 17.0 degrees (given)
Step 4: Equate the Forces
Since the block is moving at a constant speed, the applied force is balanced by the frictional force. Therefore, the horizontal force and frictional force are equal:
Horizontal Force = Frictional Force
Step 5: Solve for the Coefficient of Friction
By substituting the values from steps 2 and 3 into the equation in step 4, we can solve for the coefficient of friction:
applied force * cos(angle) = coefficient of friction * Normal Force
Substituting the known values:
8.62 N * cos(17.0 degrees) = coefficient of friction * Normal Force
Finally, rearrange the equation to solve for the coefficient of friction:
coefficient of friction = (8.62 N * cos(17.0 degrees)) / Normal Force
Substituting the values for Normal Force calculated in step 1 and solving the equation, we can determine the coefficient of friction between the block and the floor.