To determine the minimum force \( P \) required to move a block resting on a rough horizontal surface, we need to consider the forces acting on the block and the role of static friction. The coefficient of static friction, denoted as \( \mu \), plays a crucial role in this scenario.
Understanding the Forces at Play
When a force \( P \) is applied to the block, it must overcome the static frictional force to initiate movement. The static frictional force can be calculated using the formula:
- Frictional Force (f) = \( \mu \cdot N \)
Here, \( N \) is the normal force acting on the block. For a block resting on a horizontal surface, the normal force is equal to the weight of the block, which is \( mg \), where \( m \) is the mass of the block and \( g \) is the acceleration due to gravity.
Calculating the Static Friction
Thus, the maximum static frictional force can be expressed as:
- Maximum Static Friction (f_max) = \( \mu \cdot mg \)
For the block to start moving, the applied force \( P \) must be equal to or greater than this maximum static frictional force:
- Condition for Movement: \( P \geq f_{max} \)
Finding the Minimum Force \( P \)
To find the minimum value of \( P \) that will cause the block to move, we set \( P \) equal to the maximum static friction:
- Minimum Force (P) = \( \mu \cdot mg \)
Relating to the Given Options
Now, let’s analyze the options provided:
- A. \( mg \cdot \tan(\sin^{-1}(\mu)) \)
- B. \( mg \cdot \sin(\cot^{-1}(\mu)) \)
- C. \( mg \cdot \cos(\tan^{-1}(\mu)) \)
- D. \( mg \cdot \sin(\tan^{-1}(\mu)) \)
To find the correct expression, we can use the trigonometric identities related to the coefficient of friction. The coefficient of static friction \( \mu \) can be expressed in terms of angles:
- If \( \theta = \tan^{-1}(\mu) \), then \( \tan(\theta) = \mu \).
- Thus, \( \sin(\theta) = \frac{\mu}{\sqrt{1+\mu^2}} \) and \( \cos(\theta) = \frac{1}{\sqrt{1+\mu^2}} \).
From this, we can derive that:
- Using the identity for \( \tan \): \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), we can relate \( \mu \) back to the forces.
Finalizing the Answer
After evaluating the options and using the relationships between the angles and the coefficient of friction, we find that the correct expression for the minimum force \( P \) required to move the block is:
- Correct Answer: A. \( mg \cdot \tan(\sin^{-1}(\mu)) \)
This expression effectively captures the relationship between the applied force, the weight of the block, and the coefficient of static friction, allowing us to determine the minimum force needed to initiate movement.
