A 10 kg weight is on an inclined plane 10 ft long and 6 ft high. If there is no friction, what force parallel to the plane will prevent the body from sliding?
Given: 10 kg, 10 ft and 6 ft.
Solution:
Find the force parallel to the plane
Sin A = opposite/hypotenuse
Sin A = 6ft/10ft
= 0.6 ft.
Find the A of the sin.
A = sin^-1(6ft/10ft)
A = 36.87°
Find f1 and f2:
Note: m (mass)
g (gravity pulls) = 9.81 m/s^2
F1= sin (teta) x mg
= sin 36.87° x 10 kg(9.81m/s^2)
F1 = 58.86 N prevent the body from sliding
F2 = cos (teta) x mg
F2 = cos 36.87° x 10 kg(9.81m/s^2)
F2 = 78.48 N vertical mass
(6/10) 10 g = 6 g = 6*9.81 = 58.86 Newtons
To find the force parallel to the plane that will prevent the body from sliding, we need to analyze the forces acting on the weight on the inclined plane.
First, we need to break down the weight force into components. The weight force, also known as the force due to gravity, acts straight downward and can be decomposed into two components: one perpendicular to the plane and one parallel to the plane.
1. Perpendicular Component:
The weight force perpendicular to the plane can be calculated using the formula: F_perpendicular = m * g * cos(theta)
where m is the mass of the object (10 kg), g is the acceleration due to gravity (9.8 m/s^2), and theta is the angle of inclination of the plane with respect to the horizontal.
Since the given measurements are in feet, we need to convert them to meters. The length of the inclined plane is 10 ft, which is approximately 3.048 meters, and the height is 6 ft, which is approximately 1.829 meters. To find the angle of inclination, we can use the formula: theta = arctan(height/length).
theta = arctan(1.829/3.048)
theta ≈ 30.96 degrees
Plugging the values into the formula, we get:
F_perpendicular = 10 kg * 9.8 m/s^2 * cos(30.96 degrees)
F_perpendicular ≈ 84.78 N
2. Parallel Component:
The weight force parallel to the plane is responsible for the sliding motion, but since there is no friction in this case, there is no opposing force. Thus, if we want to prevent the body from sliding, we need to apply a force equal in magnitude to the parallel component of the weight force.
F_parallel = F_perpendicular * sin(theta)
F_parallel = 84.78 N * sin(30.96 degrees)
F_parallel ≈ 42.39 N
Therefore, a force of approximately 42.39 Newtons parallel to the plane will prevent the body from sliding.