Solving the Velocity of a Mass Moving Through a Loop With Friction

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To solve for the velocity of a mass moving through a loop with friction, it is essential to apply both circular motion principles and the conservation of energy. The analysis should focus on the highest point of the loop, where the normal force and gravitational force must balance to prevent free fall. The equation N + Mg = Mv^2/r illustrates the relationship between these forces, where N is the normal force, M is the mass, g is the acceleration due to gravity, v is the velocity, and r is the radius of the loop. Additionally, energy conservation must be considered to account for the work done against friction. The solution will involve deriving the velocity at the exit of the loop while factoring in these forces and energy considerations.
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hey, i have this problem where i need to use both circular motion and the conservation of energy to solve an object moving through a loop. Basically, there is friction considered and there is also a normal force when the object reaches the top of the loop. There are no values in this question, you just have to show steps in how you would find the velocity of the mass as it exits the loop. thx.
 
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Well good luck with that.
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The body is required to maintain a minimum speed to avoid free fall. The possibility of free fall is most stringent at the highest point of the loop. We, therefore, analyze the motion at the highest point with the help of the free body diagram as shown in the figure.

N+Mg = Mv^2/r
 

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