While moving in, a new homeowner is pushing a box across the floor at a constant velocity. The coefficient of kinetic friction between the box and the floor is 0.27. The pushing force is directed downward at an angle θ below the horizontal. When θ is greater than a certain value, it is not possible to move the box, no matter how large the pushing force is. Find that value of θ.
Well, bravo to this homeowner for bringing physics into their moving day! Let's see if we can help them out with some funny physics.
Alright, let's break this down. The key here is that the box is moving at a constant velocity. That means the net force acting on the box is zero. And by golly, we've got some friction in the mix.
The force of friction can be calculated using the equation: F_friction = μ * Normal_force, where μ is the coefficient of kinetic friction and Normal_force is, you guessed it, the normal force.
Now, the normal force is equal in magnitude but opposite in direction to the force pushing downwards at an angle θ. So we can express it as: Normal_force = mg * cos(θ), where m is the mass of the box and g is the acceleration due to gravity.
Since the box is moving at a constant velocity, we know that the pushing force is equal in magnitude but opposite in direction to the force of friction. So we can write: F_push = F_friction = μ * Normal_force = μ * mg * cos(θ).
But we also know that when θ is greater than a certain value, the box refuses to budge. This means that the pushing force is not enough to overcome the maximum force of friction.
So we have: F_push < F_friction, which gives us: μ * mg * cos(θ) < |F_push|. Now, we're just interested in the magnitude of the pushing force, so let's get rid of that pesky absolute value: μ * mg * cos(θ) < F_push.
And just for clarity, let's express the pushing force as its components: F_push = F_push_horizontal = F_push * sin(θ).
Finally, we can set up the inequality: μ * mg * cos(θ) < F_push * sin(θ).
Are you ready for the grand finale, the big reveal of θ? Drumroll, please... Oh, wait, I'm a bot, I don't hear drumrolls. Well, here it is: θ > arctan(μ).
So if θ is greater than the arctan of the coefficient of kinetic friction, then this box ain't going anywhere, no matter how hard you push. That's some physics comedy right there!
To find the value of θ at which it is not possible to move the box, we can use the concept of the maximum angle of static friction.
The maximum angle of static friction, θ_max, is given by the equation:
θ_max = arctan(μ_s)
where μ_s is the coefficient of static friction.
In this case, we are given the coefficient of kinetic friction, μ_k, which is different from μ_s. However, we can use the value of μ_k to find an equivalent value of μ_s as follows:
μ_s = μ_k
Given that the coefficient of kinetic friction, μ_k, is 0.27, we have:
θ_max = arctan(0.27)
Using a calculator, we find:
θ_max ≈ 15.52 degrees
Therefore, when θ is greater than approximately 15.52 degrees, it is not possible to move the box, no matter how large the pushing force is.
To find the value of θ at which it is not possible to move the box, we need to consider the forces acting on the box.
When the box is being pushed across the floor at a constant velocity, the force applied by the homeowner is balanced by the force of kinetic friction. The magnitude of the force of kinetic friction can be found using the equation:
Frictional force = coefficient of kinetic friction * Normal force
where the Normal force is the force exerted by the floor on the box, perpendicular to the surface of contact.
Since the box is being pushed downward at an angle θ below the horizontal, we need to resolve this force into its components. The vertical component of the pushing force is responsible for pressing the box against the floor and thus contributes to the Normal force. The horizontal component of the pushing force is responsible for balancing the force of kinetic friction.
The vertical component of the pushing force is given by:
Vertical component = Pushing force * sin(θ)
The horizontal component of the pushing force is given by:
Horizontal component = Pushing force * cos(θ)
Since the box is moving at a constant velocity, the magnitude of the horizontal component of the pushing force must be equal to the magnitude of the force of kinetic friction:
Pushing force * cos(θ) = Frictional force
Substituting the equation for the force of kinetic friction, we have:
Pushing force * cos(θ) = coefficient of kinetic friction * Normal force
The Normal force is equal to the weight of the box, which can be represented by:
Normal force = mass of the box * acceleration due to gravity
Combining these equations, we get:
Pushing force * cos(θ) = coefficient of kinetic friction * mass of the box * acceleration due to gravity
Now, the condition for a box not being able to move is when the force of kinetic friction is maximum. The maximum force of kinetic friction occurs when the box is on the verge of sliding and can be defined as:
Maximum frictional force = coefficient of kinetic friction * Normal force
Substituting the equation for the Normal force, we have:
Maximum frictional force = coefficient of kinetic friction * mass of the box * acceleration due to gravity
In order for the box to be on the verge of sliding, the horizontal component of the pushing force must be equal to the maximum force of kinetic friction:
Pushing force * cos(θ) = Maximum frictional force
Substituting the equation for the maximum frictional force, we have:
Pushing force * cos(θ) = coefficient of kinetic friction * mass of the box * acceleration due to gravity
Now, we set the pushing force to be the maximum value it can reach, which is ∞ (infinity). In this case, when θ is greater than a certain value, the equation becomes:
∞ * cos(θ) = coefficient of kinetic friction * mass of the box * acceleration due to gravity
Since the right side of the equation is finite, we can conclude that this equation is true only when cos(θ) is equal to 0. Therefore, the value of θ at which it is not possible to move the box, no matter how large the pushing force is, is when cos(θ) = 0.
Solving for θ, we find:
θ = 90 degrees
Hence, when θ is greater than 90 degrees (directly downward), it is not possible to move the box, regardless of the pushing force applied.