A 100-kg box is placed on a ramp. As one end of the ramp is raised, the box begins to move downward just as the angle of inclination reaches 15 degrees. What is the coefficient of static friction between box and ramp?

Wb = m * g = 100kg * 9.8N/kg = 980 N.

F1 = 980*sin15 = 253.6 N. = Force parallel to the ramp.
F2 = 980*cosi5 = 946.6 N. = Force perpendicular to the ramp.

Fs = F1 = 253.6 N. @ 15o

u = Fs/F2 = 253.6/946.6 = 0.268

Well, if the box starts moving downward at an angle of 15 degrees, it means that the force of gravity pulling the box downward is greater than the force of friction keeping it in place. So, we can say that the force of gravity is equal to the force of friction.

Using some trigonometry, we can determine that the force of gravity acting on the box is equal to 100 kg * 9.8 m/s^2 * sin(15 degrees).

Now, since the box is just about to start moving, we know that the force of friction is at its maximum, which is equal to the coefficient of static friction multiplied by the normal force (which is equal to the weight of the box).

So, if we set the force of gravity equal to the force of static friction, we can solve for the coefficient of static friction:

100 kg * 9.8 m/s^2 * sin(15 degrees) = μ * 100 kg * 9.8 m/s^2

Simplifying this equation, we find that the coefficient of static friction μ is equal to sin(15 degrees).

So, the coefficient of static friction between the box and the ramp is approximately equal to 0.2588.

And that's how you calculate the coefficient of static friction using some clever trigonometry!

To find the coefficient of static friction between the box and the ramp, we need to analyze the forces acting on the box.

1. Start by drawing a free-body diagram for the box. The weight of the box (mg) acts vertically downward, while the normal force (N) acts perpendicular to the ramp.

|<--- N --->|
| |
| |
| |
| |
|-------------|--------------------------
| <--- mg --->

2. The force due to gravity, mg, can be broken down into two components parallel and perpendicular to the ramp. The component parallel to the ramp is mg * sin(15°), and the component perpendicular to the ramp is mg * cos(15°).

N |<-- mg * sin(15°) -->|
| |
| |
| |
| |
|-------------|--------------------------
| <--- mg * cos(15°) --->

3. The box starts to move downward when the force due to gravity parallel to the ramp exceeds the maximum static friction force. Therefore, the static friction force, f_s, can be calculated as:

f_s = μ_s * N

where μ_s is the coefficient of static friction.

4. Using the component of weight perpendicular to the ramp, we can find the normal force:

N = mg * cos(15°)

5. Substitute the value of N from step 4 into the equation from step 3 to solve for μ_s:

f_s = μ_s * N
mg * sin(15°) = μ_s * mg * cos(15°)

6. Simplify the equation by canceling out the mg term:

sin(15°) = μ_s * cos(15°)

7. Solve for the coefficient of static friction:

μ_s = sin(15°) / cos(15°)

8. Use a calculator or trigonometric functions to find the numerical value of μ_s:

μ_s ≈ 0.2679

Therefore, the coefficient of static friction between the box and the ramp is approximately 0.2679.

To determine the coefficient of static friction between the box and the ramp, we need to consider the forces acting on the box.

First, let's draw a free-body diagram of the box on the inclined ramp:

```
------------
| θ |
| ↓ |
| F_norm |
| ↑ | ← F_friction
| → |
------------
```

1. The weight of the box (mg) acts vertically downward. Its magnitude is given by mg = mass (m) * acceleration due to gravity (g). In this case, m = 100 kg and g = 9.8 m/s².

2. The force exerted by the ramp on the box perpendicular to the ramp's surface is called the normal force (F_norm). It acts perpendicular to the incline and counters the weight of the box. It is equal in magnitude to the component of the weight acting perpendicular to the ramp's surface.

3. There is also a force of static friction (F_friction) acting in the opposite direction to the motion, preventing the box from sliding down the ramp. The maximum value of static friction is given by the equation F_friction = coefficient of static friction (µ_s) * F_norm.

When the box is just about to move, the static friction force reaches its maximum value. At this point, the static friction force becomes equal to the component of the weight acting parallel to the ramp's surface:

F_friction = m * g * sin(θ)

where θ is the angle of inclination, which in this case is 15 degrees.

To find µ_s, we need to divide both sides of the equation by F_norm:

µ_s = (m * g * sin(θ)) / F_norm

To calculate F_norm, we can use the component of the weight acting perpendicular to the ramp's surface:

F_norm = m * g * cos(θ)

Now we can substitute this into the equation for µ_s:

µ_s = (m * g * sin(θ)) / (m * g * cos(θ))

Simplify the equation:

µ_s = sin(θ) / cos(θ)

Finally, substitute the value of θ (15 degrees) into the equation to calculate the coefficient of static friction (µ_s).

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