A bead slides without friction around a loop the-loop. The bead is released from a height of 17.6 m from the bottom of the loop-the loop which has a radius 5 m. The acceleration of gravity is 9.8 m/s^2. Required:
How large is the normal force on it if its mass is 5g?

 N₁ = 393.96 N   and  N = 197.96 N

Explanation:

In This exercise we must use Newton's second law to find the normal force. Let's use two points the lowest and the highest of the loop

Lowest point, we write Newton's second law n for the y-axis

          N -W = m a

where the acceleration is ccentripeta

          a = v² / r

          N = W + m v² / r

          N = mg + mv² / r

we can use energy to find the speed at the bottom of the circle

starting point. Highest point where the ball is released

           Em₀ = U = m g h

lowest point. Stop curl down

            = K = ½ m v²

           Emo = Em_{f}

           m g h = ½ m v²

           v² = 2 gh

we substitute

             N = m (g + 2gh / r)

            N = mg (1 + 2h / r)

let's calculate

          N₁ = 5 9.8 (1 + 2 17.6 / 5)

          N₁ = 393.96 N

headed up

we repeat the calculation in the longest part of the loop

          -N -W = - m v₂² / r

            N = m v₂² / r - W

             N = m (v₂²/r  - g)

we seek speed with the conservation of energy

           Em₀ = U = m g h

final point. Top of circle with height 2r

              = K + U = ½ m v₂² + mg (2r)

              Em₀ =   Em_{f}

            mgh = ½ m v₂² + 2mgr

             v₂² = 2 g (h-2r)

we substitute

            N = m (2g (h-2r) / r - g)

            N = mg (2 (h-r) / r 1) = mg (2h/r  -2 -1)

             N = mg (2h/r  - 3)

            N = 5 9.8 (2 17.6 / 5 -3)

            N = 197.96 N

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