The diagram shows a cylinder of mass 20 kg lying at rest on two smooth planes inclined at angles of 40 degree and 50 degree to the horizontal. Calculate the reaction force exerted by the planes on the cylinder.
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Perform vertical and horizontal force balances (static equilibrium) and solve for the two unknown forces. Assume that both forces are normal to the smooth plane surfaces, and pass through the cylinder radii.
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The diagram shows a cylinder of mass 20 kg lying at rest on two smooth planes inclined at angles of 40 degree and 50 degree to the horizontal. Calculate the reaction force exerted by the planes on the cylinder.
I apologize, but as an AI text-based bot, I can't access or display images or links. However, I can still guide you through the process of calculating the reaction forces.
To calculate the reaction forces exerted by the planes on the cylinder, we can use the principle of resolving forces into components.
Let's denote the reaction force exerted by the plane inclined at 40 degrees as R1, and the force exerted by the plane inclined at 50 degrees as R2. Since the cylinder is at rest, the sum of the vertical components of R1 and R2 should balance the weight of the cylinder.
First, let's calculate the weight of the cylinder. The weight is the force of gravity acting on the mass and can be calculated using the formula:
Weight = mass * gravitational acceleration
Given that the mass of the cylinder is 20 kg, and the gravitational acceleration is approximately 9.8 m/s^2, we can calculate the weight:
Weight = 20 kg * 9.8 m/s^2 = 196 N
Now, we can use trigonometry to calculate the vertical components of R1 and R2.
Vertical component of R1 = R1 * sin(40°)
Vertical component of R2 = R2 * sin(50°)
The sum of these two components should equal the weight of the cylinder:
Vertical component of R1 + Vertical component of R2 = Weight
Now we have an equation with two unknowns (R1 and R2). We can solve this equation to find the reaction forces.
Let's go step by step:
1. Rearrange the equation to isolate one of the unknowns, for example, R1:
R1 = (Weight - Vertical component of R2) / sin(40°)
2. Substitute the values:
R1 = (196 N - R2 * sin(50°)) / sin(40°)
3. Substitute this value of R1 into the equation for the sum of the vertical components:
(196 N - R2 * sin(50°)) / sin(40°) + R2 * sin(50°) = Weight
Now, you can solve this equation to find the value of R2.