a force of 486 lb is necessary to keep a weight of exactly 800 lb from sliding down an inclined plane. What is the angle of inclination of the plane? Assume friction is ignored.
800 sinθ = 486
To find the angle of inclination of the plane, we can use trigonometry and the concept of the components of forces.
Let's assume that the weight is pulling down vertically, and the force applied to prevent the weight from sliding is acting parallel to the inclined plane.
We can decompose the weight force into two components: one perpendicular to the plane (normal force) and the other parallel to the plane (force applied).
Since the weight is 800 lb, the perpendicular component will also be 800 lb.
Now, we can use the trigonometric relationship between the parallel component (force applied) and the weight to find the angle of inclination (θ).
Sin(θ) = Opposite (force applied) / Hypotenuse (weight)
Sin(θ) = Force applied / Weight
Sin(θ) = 486 lb / 800 lb
Sin(θ) = 0.6075
To find the angle, we can take the inverse sine (arcsin) of this value:
θ = arcsin(0.6075)
Using a calculator, the angle of inclination (θ) is approximately 37.2 degrees.
To find the angle of inclination of the plane, we can use the concept of trigonometry. The force required to hold the weight in place is acting against the component of the weight along the inclined plane.
First, let's resolve the weight (800 lb) into its components. The weight can be broken down into two perpendicular components: the force acting along the inclined plane (parallel to the surface) and the force acting perpendicular to the inclined plane (perpendicular to the surface).
The force acting along the inclined plane can be calculated using the formula:
Force along the inclined plane = Weight * sin(angle)
where angle is the angle of inclination we want to find.
In our case, the weight is 800 lb, so the force along the inclined plane becomes:
Force along the inclined plane = 800 lb * sin(angle)
Given that the force required to prevent the weight from sliding down is 486 lb, we can set up the following equation:
486 lb = 800 lb * sin(angle)
Now, let's solve for the angle:
sin(angle) = 486 lb / 800 lb
sin(angle) ≈ 0.6075
To find the angle, we can take the inverse sine (sin^(-1)) of 0.6075:
angle ≈ sin^(-1)(0.6075)
angle ≈ 37°
Therefore, the angle of inclination of the plane is approximately 37°.