A .400 kg bead slides on a curved wire, starting from rest at point A. If the wire in problem 61 is frictionless between points A and B (down 5m) and rough between B and C (up 2m), and if the bead starts from rest at A, (a) find the speed at B. (b) If the bead comes to rest at C, find the loss in mechanical energy as it goes from B to C.
(a) (1/2)(0)^2 +(9.8m/s^2)(5m) = (1/2) vf^2 + (9.8m/s^2)(0m) = 9.90m/s
(b)?
To find the loss in mechanical energy from B to C, we need to calculate the change in mechanical energy.
The mechanical energy at point B can be calculated using the equation:
E_B = (1/2) m v_B^2 + m g h_B
where m is the mass of the bead, v_B is the speed at point B, g is the acceleration due to gravity, and h_B is the height at point B.
Given that the mass of the bead is 0.400 kg, the speed at point B is 9.90 m/s (calculated in part a), the acceleration due to gravity is 9.8 m/s^2, and the height at point B is -5 m (since it is below the starting point), we can substitute these values into the equation:
E_B = (1/2)(0.400 kg)(9.90 m/s)^2 + (0.400 kg)(9.8 m/s^2)(-5 m)
Calculating the expression gives:
E_B = 1.96 J + (-19.6 J) = -17.64 J.
The mechanical energy at point C can be calculated using the same equation:
E_C = (1/2) m v_C^2 + m g h_C
where v_C is the speed at point C and h_C is the height at point C.
Given that the bead comes to rest at point C, the speed at point C is 0 m/s. Additionally, the height at point C is +2 m (since it is above the starting point). Using these values, the equation becomes:
0 = (1/2)(0.400 kg)(0 m/s)^2 + (0.400 kg)(9.8 m/s^2)(2 m)
Simplifying the expression gives:
0 = 0 + 7.84 J = 7.84 J.
Therefore, the loss in mechanical energy from B to C can be calculated by finding the difference between E_B and E_C:
Loss in mechanical energy = E_B - E_C = (-17.64 J) - (7.84 J) = -25.48 J.
The loss in mechanical energy from B to C is therefore -25.48 J.
To solve part (b) and find the loss in mechanical energy as the bead goes from point B to point C, we need to calculate the potential energy at points B and C.
At point B, the bead is higher than at point C, so it has more potential energy. The potential energy at point B can be calculated using the formula:
Potential energy (B) = mass x gravity x height (B)
Since the mass of the bead is 0.400 kg, the acceleration due to gravity is 9.8 m/s², and the height at B is 5 m, we can substitute these values into the formula:
Potential energy (B) = 0.400 kg x 9.8 m/s² x 5 m
Potential energy (B) = 19.6 J
At point C, the bead is lower, so its potential energy is less. The potential energy at point C can be calculated in the same way:
Potential energy (C) = mass x gravity x height (C)
Given that the height at C is 2 m, we can substitute the other known values:
Potential energy (C) = 0.400 kg x 9.8 m/s² x 2 m
Potential energy (C) = 7.84 J
The loss in mechanical energy can be calculated as the difference between the potential energies at B and C:
Loss in mechanical energy = Potential energy (B) - Potential energy (C)
Loss in mechanical energy = 19.6 J - 7.84 J
Loss in mechanical energy = 11.76 J
Therefore, the loss in mechanical energy as the bead goes from point B to point C is 11.76 Joules.