A 330 kg box rests on an adjustable ramp having s = 0.25. The angle on
inclination of the ramp is increased until the box begins to slide. At what angle
will the box begin to slide?
Wb = M*g = 330 * 9.8 = 3234 N.
Fp = 3234*sin A = Force parallel to the
incline.
Fn = 3234*Cos A = Force perpendicular to
the incline.
Fs=u*Fn = 0.25 * 3234*Cos A = 809*Cos A = Force of static friction.
Fp-Fs = M*a.
3234*sin A-8o9*Cos A = M*0 = 0.
3234*sin A = 809*Cos A.
Divide both sides by Cos A:
3234*sin A/Cos A = 809.
sin A/Cos A = Tan A.
3234*Tan A = 809.
Tan A = 809/3234 = 0.25.
A = 14 Deg.
To determine the angle at which the box begins to slide, we need to consider the forces acting on the box. There are two main forces to consider: the force of gravity pulling the box downward and the force of friction acting between the box and the ramp.
The force of gravity is given by the equation F_gravity = m * g, where m is the mass of the box and g is the acceleration due to gravity (approximately 9.8 m/s^2).
The force of friction can be determined using the equation F_friction = μ * N, where μ is the coefficient of friction and N is the normal force acting on the box. The normal force is equal to the component of the force of gravity that is perpendicular to the ramp, which can be calculated as N = m * g * cos(θ), where θ is the angle of inclination.
At the point where the box begins to slide, the force of friction will be equal and opposite to the component of the force of gravity parallel to the ramp. This can be expressed as F_friction = F_gravity * sin(θ).
Combining these equations, we get:
μ * m * g * cos(θ) = m * g * sin(θ)
Simplifying:
μ * cos(θ) = sin(θ)
Dividing both sides by cos(θ), we get:
μ = tan(θ)
Now we can use the inverse tangent (arctan) function to find the angle θ:
θ = arctan(μ)
Substituting the given value of the static coefficient of friction (μ = 0.25) we can calculate the angle at which the box will begin to slide:
θ = arctan(0.25)
Using a calculator or a trigonometric table, we find that θ ≈ 14.04 degrees.
Therefore, the box will begin to slide at an angle of approximately 14.04 degrees.