A bead of mass m = 41.5 kg is released from point A which is located 5 m above the ground and slides on the frictionless track as shown in the figure. Determine the beads speed when it reaches point C which is located 2.0 m above the ground
To determine the speed of the bead when it reaches point C, we can use the principle of conservation of mechanical energy, which states that the total mechanical energy of a system remains constant as long as no external work is done.
The mechanical energy of the bead at point A is given by the sum of its potential energy and kinetic energy:
E_A = PE_A + KE_A
At point A, the bead only has potential energy, which is given by:
PE_A = m * g * h_A
where m is the mass of the bead, g is the acceleration due to gravity, and h_A is the height of point A.
Since the track is frictionless, there is no loss of mechanical energy due to friction. Therefore, the total mechanical energy at point A is equal to the total mechanical energy at point C:
E_A = E_C
The mechanical energy of the bead at point C is given by the sum of its potential energy and kinetic energy:
E_C = PE_C + KE_C
At point C, the bead has both potential energy and kinetic energy. The potential energy is given by:
PE_C = m * g * h_C
where h_C is the height of point C.
To solve for the kinetic energy at point C, we need to find the difference in potential energy between points A and C. The change in potential energy is given by:
ΔPE = PE_C - PE_A
Substituting the equations for potential energy, we have:
ΔPE = m * g * h_C - m * g * h_A
Setting the change in potential energy equal to the change in kinetic energy, we have:
ΔPE = KE_C - KE_A
Since the bead starts from rest at point A, its initial kinetic energy is zero:
KE_A = 0
Therefore, we can write:
ΔPE = KE_C - 0
Rearranging the equation, we can solve for the kinetic energy at point C:
KE_C = ΔPE
Substituting the expression for the change in potential energy, we have:
KE_C = m * g * h_C - m * g * h_A
Finally, to find the speed of the bead at point C, we can use the equation for kinetic energy:
KE_C = (1/2) * m * v^2
where v is the speed of the bead at point C.
Substituting this equation into the expression for kinetic energy at point C, we have:
(1/2) * m * v^2 = m * g * h_C - m * g * h_A
Simplifying the equation, we can solve for the speed v:
v^2 = 2 * g * (h_C - h_A)
v = sqrt(2 * g * (h_C - h_A))
Now we can plug in the given values to calculate the speed of the bead at point C.
To determine the bead's speed when it reaches point C, we can make use of the principle of conservation of mechanical energy. The total mechanical energy of the bead at any point along the track is given by the sum of its potential energy and kinetic energy.
At point A:
- The bead has only potential energy given by its height above the ground.
- The potential energy at point A is equal to the product of the bead's mass (m), the acceleration due to gravity (g), and the height of point A above the ground (5 m).
At point C:
- The bead has both potential energy and kinetic energy.
- The potential energy at point C is equal to the product of the bead's mass (m), the acceleration due to gravity (g), and the height of point C above the ground (2.0 m).
- The kinetic energy at point C is equal to the product of half the bead's mass (0.5m) and the square of its speed at point C.
Since the track is frictionless, there is no energy loss due to friction. Therefore, the total mechanical energy at point A is equal to the total mechanical energy at point C.
Mathematically, we can write the conservation of mechanical energy equation as follows:
m * g * hA = m * g * hC + 0.5m * vC^2
where:
m = mass of the bead (41.5 kg)
g = acceleration due to gravity (9.8 m/s^2)
hA = height of point A above the ground (5 m)
hC = height of point C above the ground (2.0 m)
vC = speed of the bead at point C (to be determined)
Now we can solve the equation to find the bead's speed at point C:
m * g * hA = m * g * hC + 0.5m * vC^2
41.5 kg * 9.8 m/s^2 * 5 m = 41.5 kg * 9.8 m/s^2 * 2.0 m + 0.5 * 41.5 kg * vC^2
2034.7 J = 803 J + 20.675 kg * vC^2
Subtracting 803 J from both sides:
2034.7 J - 803 J = 20.675 kg * vC^2
1231.7 J = 20.675 kg * vC^2
Dividing both sides by 20.675 kg:
59.6 m^2/s^2 = vC^2
Taking the square root of both sides:
vC = √(59.6 m^2/s^2)
vC ≈ 7.72 m/s
Therefore, the bead's speed when it reaches point C is approximately 7.72 m/s.