Math Problem Statement
A block is at rest on the incline shown in the figure. The coefficients of static and kinetic friction are µs = 0.7 and µk = 0.59, respectively. The acceleration of gravity is 9.8 m/s 2 . 31 kg µ 31◦ What is the frictional force acting on the 31 kg mass? Answer in units of N.
Solution
To find the frictional force acting on the 31 kg mass, we first need to determine whether the block is moving or at rest. This will be based on the static friction.
Step 1: Calculate the gravitational force components
The gravitational force acting on the block is:
The component of this force parallel to the incline () is given by:
The component of the gravitational force perpendicular to the incline () is:
Step 2: Maximum static friction force
The maximum static friction force is given by:
Step 3: Compare the forces
The frictional force needs to balance the parallel component of the gravitational force () to keep the block at rest. Since the maximum static friction force (182.21 N) is greater than (156.45 N), the block does not move. Therefore, the static friction force is equal to the parallel component of gravity:
Thus, the frictional force acting on the 31 kg block is 156.45 N.
Would you like further details or have any questions?
Here are five related questions to expand on this topic:
- How does the angle of the incline affect the frictional force?
- What happens if the angle is increased beyond a certain point?
- How would the frictional force change if the mass of the block increased?
- What would the frictional force be if the block started moving?
- How does kinetic friction differ from static friction in this context?
Tip: Remember that static friction adjusts up to its maximum value to prevent motion, while kinetic friction has a constant value once motion starts.
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Math Problem Analysis
Mathematical Concepts
Physics
Static Friction
Inclined Plane
Trigonometry
Formulas
Gravitational force: F_g = m * g
Parallel component of gravitational force: F_parallel = F_g * sin(θ)
Perpendicular component of gravitational force: F_perpendicular = F_g * cos(θ)
Maximum static friction: f_s_max = µ_s * F_perpendicular
Theorems
Newton's Laws of Motion
Suitable Grade Level
Grades 11-12
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