An experiment is to be conducted to determine the coefficient of kinetic friction between a block and a flat plank of wood. The plank is tilted at an angle θ, and the block slides down the plank. It is observed that the block of mass m slides at a constant speed.
What quantities must be measured in order to calculate the coefficient of friction?
The Correct Answer and Explanation is:
To determine the coefficient of kinetic friction (( \mu_k )) between the block and the plank, the angle of inclination (( \theta )) of the plank must be measured. Since the block is sliding at a constant speed, we know that the forces along the incline are balanced, allowing us to calculate ( \mu_k ) based solely on ( \theta ) and the gravitational force.
Here’s how it works:
When the block slides down the inclined plane at a constant speed, it indicates that the net force acting along the direction of the incline is zero. The gravitational force pulling the block down the incline (( mg \sin \theta )) is exactly balanced by the kinetic frictional force (( f_k )), which opposes the motion. The kinetic frictional force is given by:
[
f_k = \mu_k \cdot N
]
where ( N ) is the normal force exerted by the inclined plane on the block. On an incline, the normal force is less than the full gravitational force and is given by:
[
N = mg \cos \theta
]
Since the block is moving at a constant velocity, the forces are in equilibrium along the incline:
[
mg \sin \theta = \mu_k \cdot mg \cos \theta
]
By canceling ( mg ) from both sides, we get:
[
\mu_k = \tan \theta
]
Thus, the only quantity needed to calculate ( \mu_k ) is the angle of inclination (( \theta )). When ( \theta ) is measured accurately, the coefficient of kinetic friction can be determined directly as ( \tan \theta ). This setup makes the experiment simpler, as the mass of the block or the gravitational constant is not required, given they cancel out in the equations.
In summary, measuring the angle ( \theta ) of the plank is sufficient to determine ( \mu_k ) when the block slides at a constant speed. This approach leverages the balance of forces along the incline and simplifies the calculation by using trigonometric relationships.