A block is at rest on an inclined plane.
63 kg µs = 0.47
èc
Find the critical angle, èc, at which the block just begins to slide. Answer in units of ◦
θ = tan⁻¹μ
M*g = 63 * 9.8 = 617.4 N. = Wt. of block
Fp = Fs @ critical angle.
Mg*sinA = us*Mg*CosA
Divide both sides by Mg:
sin A = us*Cos A
Divide both sides by Cos A:
sin A/Cos A = us = 0.47 = Tan A
A = 25.2o = The critical angle.
Fp = 262.6
Why did the block bring a flashlight to the inclined plane?
Because it wanted to shed some light on its critical angle!
To find the critical angle at which the block just begins to slide, we can use the formula for the maximum static friction force:
f_max = µs * N
Where:
- f_max is the maximum static friction force
- µs is the coefficient of static friction
- N is the normal force
In this case, the normal force, N, can be calculated by decomposing the weight of the block into two components: one perpendicular to the inclined plane (N), and one parallel to the inclined plane (mg * sin(θ)):
N = mg * cos(θ)
Where:
- m is the mass of the block (63 kg)
- g is the acceleration due to gravity (9.8 m/s^2)
- θ is the angle of inclination of the plane
Since the block is at rest, the maximum static friction force is equal to the component of the weight of the block parallel to the inclined plane:
f_max = mg * sin(θ)
Setting these two expressions for f_max equal to each other:
mg * sin(θ) = µs * mg * cos(θ)
The mass (m) and the acceleration due to gravity (g) cancel out, yielding:
sin(θ) = µs * cos(θ)
To find the critical angle at which the block just begins to slide, we need to find the value of θ that satisfies this equation. However, since there is no direct algebraic solution for this equation, we can use numerical methods or a table of trigonometric values to find the answer.
Using a scientific calculator or a table of trigonometric values, we can find that the critical angle is approximately 28.9 degrees.
To find the critical angle (èc) at which the block just begins to slide on an inclined plane, we need to take into account the weight of the block and the force of friction acting on it.
The force of friction can be calculated using the equation:
f = µs * N
Where:
- f is the force of friction,
- µs is the coefficient of static friction, given as 0.47 in this case, and
- N is the normal force exerted on the block by the inclined plane.
The normal force, N, is equal to the weight of the block, which can be found using the equation:
W = m * g
Where:
- W is the weight of the block,
- m is the mass of the block, given as 63 kg in this case, and
- g is the acceleration due to gravity, which is approximately 9.8 m/s².
Substituting the values, we find:
W = 63 kg * 9.8 m/s²
W ≈ 617.4 N
With the weight determined, we can now calculate the force of friction:
f = 0.47 * 617.4 N
f ≈ 290.078 N
The angle at which the block just begins to slide, the critical angle (èc), can be determined using the formula:
tan(èc) = µs
Rearranging the equation, we get:
èc = tan⁻¹(µs)
Substituting the coefficient of static friction, we find:
èc = tan⁻¹(0.47)
Evaluating this in a calculator, we get:
èc ≈ 25.36°
Therefore, the critical angle (èc) at which the block just begins to slide on the inclined plane is approximately 25.36°.