A block with a mass of m=2.07kg
 rests on a wooden plank. The coefficient of static friction between the block and the plank is μs=0.79
. One end of the board is attached to a hinge so that the other end can be lifted forming an angle, θ
, with respect to the ground. Assume the x
 axis is parallel to the surface of the plank, as shown in the figure.Write an expression for the maximum angle, θmax, that the board can make with respect to the horizontal before the block starts moving. (Write in terms of the given parameters and variables available in the palette.)

To determine the maximum angle \(\theta_{\text{max}}\) at which the block will start sliding, we need to consider the forces acting on the block and how they relate to the frictional force.

### Forces Analysis
1. **Gravitational Force (\(mg\))**: Acts vertically downward, where \( m = 2.07 \, \text{kg} \) and \( g \) is the acceleration due to gravity.
2. **Normal Force (\(N\))**: Perpendicular to the surface of the inclined plane.
3. **Frictional Force (\(f_s\))**: Acts parallel to the surface, opposing motion, with a maximum value of \( f_s = \mu_s N \), where \( \mu_s = 0.79 \).
4. **Component of Gravity Along the Plane (\(mg \sin \theta\))**: Acts down the slope, causing the block to slide.
5. **Component of Gravity Perpendicular to the Plane (\(mg \cos \theta\))**: Balances the normal force, \( N \).

### Conditions for Maximum Angle
The block will start moving when the component of gravity parallel to the slope (\(mg \sin \theta\)) equals the maximum static frictional force (\(f_s\)):

\[
mg \sin \theta = \mu_s N
\]

Since \( N = mg \cos \theta \), we can substitute it in:

\[
mg \sin \theta = \mu_s mg \cos \theta
\]

Simplifying by canceling \( mg \) from both sides:

\[
\sin \theta = \mu_s \cos \theta
\]

Dividing both sides by \( \cos \theta \):

\[
\tan \theta = \mu_s
\]

Thus, the maximum angle \(\theta_{\text{max}}\) is:

\[
\theta_{\text{max}} = \tan^{-1}(\mu_s)
\]

### Final Expression
\[
\theta_{\text{max}} = \tan^{-1}(0.79)
\]

If you need a numerical value for this angle, let me know!

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Would you like any further details or have specific questions on this?

### Related Questions
1. How does increasing the coefficient of static friction affect \(\theta_{\text{max}}\)?
2. What is the expression for the normal force \(N\) at an arbitrary angle \(\theta\)?
3. How would the solution change if the plank were moving, involving kinetic friction?
4. How does changing the mass \(m\) of the block influence the outcome?
5. What are the equilibrium conditions for the block before it starts moving?

**Tip:** Understanding the breakdown of forces into parallel and perpendicular components helps solve many physics problems involving inclines.

A block with a mass of m=2.07kg rests on a wooden plank. The coefficient of static friction between the block and the plank is μs=0.79. One end of the board is attached to a hinge so that the other end can be lifted forming an angle, θ, with respect to the ground. Write an expression for the maximum angle, θmax, that the board can make with respect to the horizontal before the block starts moving.

Learn how to determine the maximum angle (θmax) at which a block will begin to move on an inclined plane with static friction. The problem involves key physics concepts such as trigonometry, static friction, and inclined plane forces. This solution is ideal for high school students studying physics.

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