Math Problem Statement

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=0.59 . 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. Part (b) With rotated coordinate axes that were presented, find an expression for the component of the force of gravity in the y direction and perpendicular to the plank, in terms of given quantities and variables available in the palette.

Solution

To find the component of the force of gravity in the yy-direction (perpendicular to the plank), we'll decompose the gravitational force into components along the rotated coordinate axes.

Definitions:

  1. mm: Mass of the block = 2.07 kg.
  2. μs\mu_s: Coefficient of static friction = 0.59.
  3. θ\theta: Angle of the plank with respect to the ground.
  4. gg: Acceleration due to gravity = 9.8 m/s2^2.

Gravity Force:

The gravitational force acting on the block is Fg=mgF_g = mg. This force acts vertically downward.

Rotated Coordinate System:

  • xx-axis: Parallel to the plank.
  • yy-axis: Perpendicular to the plank.

Components of Gravity:

  1. Along the yy-axis (perpendicular to the plank): Fgy=mgcosθF_{gy} = mg \cos \theta

  2. Along the xx-axis (parallel to the plank): Fgx=mgsinθF_{gx} = mg \sin \theta

Explanation:

  • The component FgyF_{gy} is responsible for the normal force exerted by the plank on the block.
  • The component FgxF_{gx} is responsible for trying to move the block down the plank due to gravity.

In terms of given variables:

Fgy=mgcosθF_{gy} = mg \cos \theta

where:

  • mm is the mass of the block (2.07 kg).
  • gg is the acceleration due to gravity (9.8 m/s2^2).
  • θ\theta is the angle between the plank and the horizontal.

Do you need further details or have any questions?

Here are 5 related questions to explore the concept further:

  1. What is the normal force acting on the block if the angle θ\theta is 30°?
  2. How does the frictional force change as the angle θ\theta increases?
  3. At what angle θ\theta does the block start to slide down the plank?
  4. How would the static friction coefficient affect the critical angle at which the block begins to slide?
  5. What is the component of the gravitational force along the xx-axis at θ=45\theta = 45^\circ?

Tip: Always resolve forces into components along the axes that simplify your problem, especially when dealing with inclined planes or non-standard coordinate systems.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Forces in Rotated Coordinate Systems
Newton's Laws of Motion

Formulas

F_g = mg
F_gy = mg cos(θ)
F_gx = mg sin(θ)

Theorems

Newton's Second Law of Motion

Suitable Grade Level

Grades 10-12