You push down on a book of inertia m resting on a table with a force directed at an angle θ away from vertical. The coefficient of static friction between book and table is μs. If θ is not larger than some critical value, you cannot get the book to slide no matter how hard you push.
What is that critical value?
answer is in terms of μs
Im not sure where to start
I had this same problem... θ > arctan(μs) is the answer, you're welcome fam.
To determine the critical value of the angle θ, we need to analyze the forces acting on the book and calculate the maximum angle at which the static friction force can prevent the book from sliding.
Let's consider the different forces acting on the book:
1. The weight force (mg), acting vertically downward.
2. The normal force (N), exerted by the table, which is equal to the weight force since the book is at rest.
3. The force of static friction (fs), exerted by the table, which opposes the applied force and prevents the book from sliding.
We can break down the applied force into its horizontal and vertical components:
1. The vertical component of the applied force equals F_y = F × sin(θ).
2. The horizontal component of the applied force equals F_x = F × cos(θ).
Since the book is at rest, the maximum static friction force is fs = μs × N = μs × mg, and it acts parallel to the table surface.
To prevent the book from sliding, the horizontal component of the applied force (F_x) should be less than or equal to the static friction force (fs):
F_x ≤ fs
=> F × cos(θ) ≤ μs × mg
We can rearrange this equation to solve for the critical angle (θc):
cos(θc) ≤ μs × (mg/F)
=> θc ≤ arccos(μs × (mg/F))
Therefore, the critical value for the angle θ is given by:
θc = arccos(μs × (mg/F))
Note: The critical angle will depend on the ratio between the static friction coefficient (μs) and the ratio of the weight force to the applied force (mg/F).
To determine the critical value of θ, we need to consider the forces acting on the book and analyze the conditions for it to start sliding. Let's break down the forces involved:
1. The weight force (mg) acting vertically downward on the book, where m is the mass of the book and g is the acceleration due to gravity.
2. The normal force (N) exerted by the table on the book, which is equal in magnitude but opposite in direction to the weight force (N = mg).
3. The force you apply to the book (F) at an angle θ away from the vertical.
4. The static frictional force (fs) opposing the applied force, which prevents the book from moving until it reaches its maximum value. The magnitude of static friction is given by fs = μsN, where μs is the coefficient of static friction.
For the book to remain in equilibrium (not sliding), the sum of the forces in both the horizontal and vertical directions should be zero. Since only the horizontal forces are relevant in this case, we'll focus on those.
In the horizontal direction:
The applied force (F) can be resolved into two components: F_horizontal = F cos θ and F_vertical = F sin θ.
The static frictional force (fs) will act in the opposite direction to counterbalance the horizontal component of the applied force:
fs = - F_horizontal = - F cos θ.
Since the book is in equilibrium, the static frictional force (fs) must reach its maximum value (fs_max) before the book starts sliding. Therefore, we have:
|fs_max| = |μsN| = |μs(mg)| = F_horizontal.
In order to solve for the critical value of θ, we need to determine the angle at which the horizontal component of the applied force (F_horizontal) equals the maximum static frictional force (fs_max). This occurs when |F_horizontal| = |fs_max|, so we can set up the equation:
|F cos θ| = |μs(mg)|.
From this equation, we can solve for the critical value of θ:
cos θ = μs(mg) / F.
Finally, taking the inverse cosine of both sides, we find:
θ = arccos(μs(mg) / F).
Therefore, the critical value of θ is given by θ = arccos(μs(mg) / F), where μs is the coefficient of static friction, m is the mass of the book, g is the acceleration due to gravity, and F is the magnitude of the applied force directed at an angle θ away from vertical.