The diagram shows a cylinder of 8kg lying at rest on two smooth planes inclined at angles of 40 degrees and 50 degrees to the horizontal.
Calculate the reaction exerted by each plane on the cylinder.
Thanks in advance
To calculate the reaction exerted by each plane on the cylinder, we need to consider the forces acting on the cylinder and apply the principles of equilibrium.
We have a cylinder lying at rest on two inclined planes. Since the planes are smooth, there is no frictional force acting on the cylinder.
First, let's resolve the weight of the cylinder into components parallel and perpendicular to each plane.
The weight of the cylinder can be determined using the formula:
Weight = mass × acceleration due to gravity
Weight = 8 kg × 9.8 m/s^2
Weight = 78.4 N
To find the component of the weight parallel to each plane (force acting down the incline), we multiply the weight by the sine of the angle of inclination.
Component of weight parallel to the 40-degree inclined plane:
Force_parallel_40 = Weight × sin(40°)
Force_parallel_40 = 78.4 N × sin(40°)
Force_parallel_40 ≈ 50.3 N
Component of weight parallel to the 50-degree inclined plane:
Force_parallel_50 = Weight × sin(50°)
Force_parallel_50 = 78.4 N × sin(50°)
Force_parallel_50 ≈ 59.9 N
Since the cylinder is at rest, the net force acting along the perpendicular direction (normal to the incline) should be zero. This means that the sum of the reactions exerted by the two inclined planes should balance the component of weight perpendicular to each plane.
The component of weight perpendicular to each plane can be found by multiplying the weight by the cosine of the angle of inclination.
Component of weight perpendicular to the 40-degree inclined plane:
Weight_perpendicular_40 = Weight × cos(40°)
Weight_perpendicular_40 = 78.4 N × cos(40°)
Weight_perpendicular_40 ≈ 59.9 N
Component of weight perpendicular to the 50-degree inclined plane:
Weight_perpendicular_50 = Weight × cos(50°)
Weight_perpendicular_50 = 78.4 N × cos(50°)
Weight_perpendicular_50 ≈ 50.3 N
Since the cylinder is at rest, the reactions exerted by the inclined planes in the perpendicular direction should be equal to the component of weight perpendicular to each plane.
Reaction exerted by the 40-degree inclined plane = Weight_perpendicular_40 ≈ 59.9 N
Reaction exerted by the 50-degree inclined plane = Weight_perpendicular_50 ≈ 50.3 N
Therefore, the reaction exerted by the 40-degree inclined plane on the cylinder is approximately 59.9 N, and the reaction exerted by the 50-degree inclined plane on the cylinder is approximately 50.3 N.