A box of mass 14.0 kg sits on a horizontal steel ramp 4.8 m long with a coefficient of static friction of 0.40 between the ramp and the box. The end of the ramp is slowly lifted until the box begins to slide down the ramp. What angle does the ramp make with the ground when this happens?

Once the box begins to slide down the ramp, the coefficient of kinetic friction is 0.26. How fast will the box be moving once it reaches the bottom of the ramp?

1.

x: ma=mg•sin α –F(fr)
y: 0=-mg•cos α +N
ma=0
0= mg•sin α-μN=
= mg•sin α-μ•mg•cos α
tan α = μ =0.4
α=arctan0.4 =21°
2.
ma=mg•sin α – μ₁mg•cos α
a=g•(sin α – μ₁•cos α) =9.8(sin22°-0.26•cos22°)=…
s=v²/2a
v=sqrt(2as)=…

To find the angle the ramp makes with the ground when the box begins to slide, we can use the formula for the coefficient of static friction:

μs = tan(θ)

where μs is the coefficient of static friction and θ is the angle the ramp makes with the ground.

Given that the coefficient of static friction is 0.40, we can solve for θ:

0.40 = tan(θ)

Taking the inverse tangent (arctan) of both sides, we find:

θ = arctan(0.40)

Using a calculator, the angle is approximately 21.8 degrees.

Now, to find the speed of the box once it reaches the bottom of the ramp, we can use the principle of conservation of mechanical energy. The initial potential energy at the top of the ramp is equal to the final kinetic energy at the bottom of the ramp.

The initial potential energy (U_i) is given by:

U_i = m * g * h

where m is the mass of the box, g is the acceleration due to gravity, and h is the vertical height of the ramp.

The final kinetic energy (K_f) is given by:

K_f = (1/2) * m * v^2

where m is the mass of the box and v is the final velocity.

Since there is no vertical displacement from the top of the ramp to the bottom, the initial potential energy is equal to zero (U_i = 0). Thus, we have:

0 = (1/2) * m * v^2

Simplifying the equation, we find:

v^2 = 0

Therefore, the box will be stationary (not moving) once it reaches the bottom of the ramp.

So, the box will not have any speed when it reaches the bottom of the ramp.

To determine the angle at which the box begins to slide down the ramp, we can use the coefficient of static friction and the gravitational force acting on the box.

1. Calculate the gravitational force acting on the box:
The gravitational force can be calculated using the formula: Fg = m * g, where m is the mass of the box and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Fg = 14.0 kg * 9.8 m/s^2
Fg = 137.2 N

2. Calculate the maximum static friction force:
The maximum static friction force can be determined by multiplying the coefficient of static friction by the gravitational force. Since the box is at the point of sliding, the static friction force must be equal to the maximum static friction force.
Ffs_max = μs * Fg
Ffs_max = 0.40 * 137.2 N
Ffs_max = 54.88 N

3. Find the component of the gravitational force parallel to the ramp:
Since the ramp is angled, we need to find the component of the gravitational force that acts parallel to the ramp. This component is given by: Fg_parallel = Fg * sin(θ), where θ is the angle of the ramp with the ground.
Fg_parallel = 137.2 N * sin(θ)

4. Equate the maximum static friction force to the parallel component of the gravitational force:
At the point where the box begins to slide, the maximum static friction force is equal to the component of the gravitational force parallel to the ramp.
54.88 N = 137.2 N * sin(θ)

5. Solve for θ by isolating sin(θ):
sin(θ) = 54.88 N / 137.2 N
sin(θ) ≈ 0.4
θ ≈ sin^(-1)(0.4)
θ ≈ 23.6 degrees

Therefore, the angle the ramp makes with the ground when the box begins to slide is approximately 23.6 degrees.

Now let's move on to the second part of the question: determining the speed of the box at the bottom of the ramp.

6. Calculate the net force acting on the box when it slides down the ramp:
The net force is the difference between the gravitational force acting on the box and the kinetic friction force.
Fnet = Fg_parallel - Ffk

7. Calculate the kinetic friction force:
The kinetic friction force can be determined by multiplying the coefficient of kinetic friction by the gravitational force.
Ffk = μk * Fg
Ffk = 0.26 * 137.2 N
Ffk = 35.672 N

8. Calculate the net force:
Fnet = 137.2 N * sin(θ) - 35.672 N
Fnet = 137.2 N * sin(23.6 degrees) - 35.672 N

9. Calculate the acceleration:
The acceleration can be determined by dividing the net force by the mass of the box.
a = Fnet / m
a = (137.2 N * sin(23.6 degrees) - 35.672 N) / 14.0 kg

10. Use kinematic equations to find the final velocity:
We can use the kinematic equation v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity (which is 0), a is the acceleration, and s is the distance traveled down the ramp (4.8 m).
v^2 = 0 + 2 * (137.2 N * sin(23.6 degrees) - 35.672 N) / 14.0 kg * 4.8 m

11. Solve for v:
v = √(2 * (137.2 N * sin(23.6 degrees) - 35.672 N) / 14.0 kg * 4.8 m)

Calculating the exact value of v will give the speed of the box at the bottom of the ramp.