A block is resting at the top of a rough incline with coefficients of kinetic and static friction of 0.25 and 0.45 respectively. The angle of the incline is gradually increased until the block begins to slip. Calculate the angle that the block begins to slip.
To calculate the angle at which the block begins to slip, we need to consider the forces acting on the block. These forces include the gravitational force pulling the block downhill, the normal force exerted by the inclined plane, and the friction force opposing the motion.
Let's break down the problem step by step:
1. Begin by drawing a free-body diagram to visualize the forces acting on the block.
- The weight of the block (mg) acts vertically downward.
- The normal force (N) acts perpendicular to the incline.
- The friction force (F) acts parallel to the incline in the opposite direction of the impending motion.
2. Decompose the weight force into its components.
- The weight force can be split into two components: mg sin(θ) and mg cos(θ), where θ is the angle of the incline.
3. Calculate the normal force.
- The normal force (N) is equal in magnitude but opposite in direction to the vertical component of the weight force, so N = mg cos(θ).
4. Determine the maximum friction force.
- The maximum friction force (Fmax) can be found by multiplying the coefficient of static friction (μs) by the normal force. In this case, Fmax = μs * N.
5. Calculate the critical angle of the incline.
- The critical angle (θc) is the angle at which the maximum friction force is equal to the horizontal component of the weight force.
- In other words, Fmax = mg sin(θc).
- Substitute Fmax = μs * N and N = mg cos(θ) into the equation: μs * mg cos(θ) = mg sin(θc).
- Simplify the equation: μs / sin(θc) = cos(θ).
- Rearrange it to: sin(θc) = μs / cos(θ).
- Take the inverse sine of both sides to find the angle θc: θc = arcsin(μs / cos(θ)).
6. Substitute the given coefficients of static friction.
- In this case, the coefficient of static friction is given as 0.45 (μs = 0.45), so use this value to calculate the critical angle (θc).
By following these steps, you can determine the angle at which the block begins to slip by solving the equation θc = arcsin(μs / cos(θ)).