A 0.377 kg bead slides on a curved wire, starting from rest at point A(4.8 meters) as shown in the figure.
The acceleration of gravity is 9.8 m/s2 .
A. If the wire is frictionless, find the speed of the bead at B(0 Meters).
Answer in units of m/s
B. Find the speed of the bead at C(1.57 meters).
Answer in units of m/s
To solve this problem, we can use conservation of mechanical energy. The mechanical energy at point A is equal to the mechanical energy at any other point along the wire.
The mechanical energy of the bead consists of two components: kinetic energy (KE) and potential energy (PE).
1. Speed of the bead at point B:
At point B, the bead is at a lower height compared to point A, so it has lost some potential energy. However, the wire is frictionless, so there is no work done against friction, and the mechanical energy is conserved.
The mechanical energy at point A can be written as:
EA = KEA + PEA
EA = 0.5 * mass * velocityA^2 + mass * g * heightA
Since the bead starts from rest at point A, its initial velocity (velocityA) is 0. Therefore, the kinetic energy at point A is also 0.
EA = PEA
0.5 * mass * velocityA^2 + mass * g * heightA = mass * g * heightB
Simplifying the equation:
0.5 * velocityA^2 = g * (heightB - heightA)
To find the velocity at point B, we can rearrange the equation as follows:
velocityA^2 = 2 * g * (heightB - heightA)
velocityA = √(2 * g * (heightB - heightA))
Substituting the given values:
mass = 0.377 kg
g = 9.8 m/s^2
heightA = 4.8 m
heightB = 0 m
velocityA = √(2 * 9.8 * (0 - 4.8))
velocityA = √(-2 * 9.8 * 4.8)
velocityA = √(-93.12)
Since the velocity cannot be negative, there must be an error in the given values or the problem setup.
2. Speed of the bead at point C:
To find the speed at point C, we can use the same principle of conservation of mechanical energy. The mechanical energy at point C will be equal to the mechanical energy at point A.
The equation is given by:
EA = EC
0.5 * mass * velocityA^2 + mass * g * heightA = 0.5 * mass * velocityC^2 + mass * g * heightC
Since the bead starts from rest at point A, we know that velocityA is 0. Therefore, the equation becomes:
0 + mass * g * heightA = 0.5 * mass * velocityC^2 + mass * g * heightC
Simplifying the equation:
mass * g * heightA = mass * g * heightC
Canceling out mass and g, we get:
heightA = heightC
Therefore, the height at point C is the same as the height at point A.
Given that heightA = 4.8 meters, the speed of the bead at point C is 0 m/s.
To find the speed of the bead at point B, we can use conservation of energy. The total mechanical energy at point A is equal to the total mechanical energy at point B.
Since the wire is frictionless, the only force acting on the bead is the force due to gravity. The work done by gravity is equal to the change in potential energy.
Let's calculate the potential energy at point A.
Potential energy at A = m * g * h
where m = mass of the bead = 0.377 kg,
g = acceleration due to gravity = 9.8 m/s^2,
h = height above the reference point (B) = 4.8 m
Potential energy at A = 0.377 kg * 9.8 m/s^2 * 4.8 m
Now, let's calculate the potential energy at point B.
Potential energy at B = m * g * h
where h = height above the reference point (B) = 0 m
Potential energy at B = 0.377 kg * 9.8 m/s^2 * 0 m
Since the total mechanical energy is conserved, the potential energy at A is equal to the kinetic energy at B.
Kinetic energy at B = Potential energy at A
0.5 * m * v^2 = 0.377 kg * 9.8 m/s^2 * 4.8 m
Simplifying the equation:
0.5 * v^2 = 0.377 kg * 9.8 m/s^2 * 4.8 m
Now, let's solve for v.
v^2 = (0.377 kg * 9.8 m/s^2 * 4.8 m) / 0.5
v^2 = 18.70944 kg * m^2/s^2
Taking the square root of both sides:
v = √(18.70944 kg * m^2/s^2)
v ≈ 4.326 m/s
Therefore, the speed of the bead at point B is approximately 4.326 m/s.
To find the speed of the bead at point C, we can use the conservation of mechanical energy again.
Let's calculate the potential energy at point C.
Potential energy at C = m * g * h
where h = height above the reference point (B) = 1.57 m
Potential energy at C = 0.377 kg * 9.8 m/s^2 * 1.57 m
Since the total mechanical energy is conserved, the potential energy at A is equal to the kinetic energy at C.
Kinetic energy at C = Potential energy at A
0.5 * m * v^2 = 0.377 kg * 9.8 m/s^2 * 4.8 m
Simplifying the equation:
0.5 * v^2 = 0.377 kg * 9.8 m/s^2 * 1.57 m
Now, let's solve for v.
v^2 = (0.377 kg * 9.8 m/s^2 * 1.57 m) / 0.5
v^2 = 5.683283 kg * m^2/s^2
Taking the square root of both sides:
v = √(5.683283 kg * m^2/s^2)
v ≈ 2.38 m/s
Therefore, the speed of the bead at point C is approximately 2.38 m/s.