A uniform board is leaning against a smooth vertical wall. The board is at an angle above the horizontal ground. The coefficient of static friction between the ground and the lower end of the board is 0.710. Find the smallest value for the angle , such that the lower end of the board does not slide along the ground.
A = angle of board up
force up = weight of board = w
friction force max = .7 w
equal and opposite horizontal force on wall = .7 w
length of board = L
weight of board down = w at L/2
moment of that weight about foot of board = w(L/2) cos A
moment of horizontal wall force about foot of board = .7 w (L/2) sin A
so
.7 sin A = cos A
sin A/cos A = 1/.7 = 1.43 = tan A
so A = 55 degrees
To find the smallest value for the angle θ such that the lower end of the board does not slide along the ground, we need to consider the forces acting on the board.
1. Draw a free body diagram of the forces acting on the board. Label the forces.
- The weight of the board, acting straight downward, can be represented by mg, where m is the mass of the board and g is the acceleration due to gravity.
- The normal force, which is perpendicular to the ground, can be represented by N.
- The static frictional force, which opposes the tendency of the board to slide along the ground, can be represented by fs.
2. Write down the equation of equilibrium in the vertical direction.
The sum of the vertical forces must equal zero.
N - mg*cos(θ) = 0
Since the board's weight acts straight downward, the vertical component of the weight is mg*cos(θ).
3. Write down the equation of equilibrium in the horizontal direction.
The horizontal component of the static frictional force must equal the horizontal component of the weight.
fs = mg*sin(θ)
Since the board is at equilibrium, the horizontal component of the static frictional force, fs, must be equal to the horizontal component of the weight, mg*sin(θ).
4. Substitute the expression for fs from step 3 into the equation in step 2.
N - mg*cos(θ) = mg*sin(θ)
5. Solve the equation for the angle θ.
N = mg*cos(θ) + mg*sin(θ)
N = mg*(cos(θ) + sin(θ))
Divide both sides by mg:
N/mg = cos(θ) + sin(θ)
Simplify the equation by noting that cos(θ) = sin(π/2 - θ):
N/mg = sin(π/2 - θ) + sin(θ)
6. Identify the relationship between the expression N/mg and the coefficient of static friction.
The coefficient of static friction, μ, is the ratio of fs to N:
μ = fs/N
Rearranging the equation, we have:
N = fs/μ
Substitute this expression for N in the equation from step 5:
fs/μmg = sin(π/2 - θ) + sin(θ)
Simplify:
fs/(μmg) = sin(π/2 - θ) + sin(θ)
7. Substitute the value of the coefficient of static friction, μ = 0.710, into the equation from step 6, and solve for the angle θ.
fs/(0.710mg) = sin(π/2 - θ) + sin(θ)
The smallest value for the angle θ can be found by setting the left-hand side of the equation to its maximum value of 1:
1 = sin(π/2 - θ) + sin(θ)
The maximum value of the sum of two sine functions is 1 when the angles are equal:
π/2 - θ = θ
Rearrange the equation and solve for θ:
π/2 = 2θ
θ = π/4
Therefore, the smallest value for the angle θ such that the lower end of the board does not slide along the ground is π/4 radians or 45 degrees.
To find the smallest value for the angle θ, we need to analyze the forces acting on the board. Let's break it down step by step:
1. Draw a free-body diagram: Draw the board as a right triangle leaning against the wall. Label the angle between the board and the ground as θ.
2. Identify the forces: There are three forces acting on the board:
a. Weight (W): The force exerted by the board due to gravity. It acts in a vertically downward direction and has a magnitude of mg, where m is the mass of the board and g is the acceleration due to gravity.
b. Normal force (N): The force exerted by the ground on the bottom end of the board in a direction perpendicular to the ground. It balances the weight of the board and acts in the upward direction.
c. Frictional force (f): The force exerted by the ground on the bottom end of the board parallel to the ground. It acts in the opposite direction to the motion or tendency of motion and prevents sliding. Its magnitude is given by f = μN, where μ is the coefficient of static friction between the ground and the board.
3. Equilibrium condition: In order for the lower end of the board not to slide along the ground, the sum of the forces in both the horizontal and vertical direction should be zero.
4. Vertical equilibrium:
∑Fy = N - W sinθ = 0
N = W sinθ
5. Horizontal equilibrium:
∑Fx = f - W cosθ = 0
f = W cosθ
6. Substitute the value of f in terms of N:
W cosθ = μN
7. Substitute the value of N from the vertical equilibrium equation:
W cosθ = μW sinθ
cosθ = μ sinθ
tanθ = 1/μ
8. Find the inverse of the tangent function:
θ = arctan(1/μ)
9. Substitute the given value for the coefficient of static friction:
θ = arctan(1/0.710)
10. Use a calculator to find the value of θ:
θ ≈ 39.80 degrees
Therefore, the smallest value for the angle θ such that the lower end of the board does not slide along the ground is approximately 39.80 degrees.