The masses are each initially 1.8m above the ground, and the massless frictionless pulley is 4.8m above the ground. (a) calculate the velocity of the lighter mass at the instance the heavier mass hits the ground. (b) find the maximum height reached by the lighter object (from the ground level) after the system is released. Use the energy method to find the answers.
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To solve this problem using the energy method, we need to consider the conservation of mechanical energy.
(a) To find the velocity of the lighter mass at the instance the heavier mass hits the ground, we can equate the initial potential energy of the system to the final kinetic energy of the lighter mass.
Let's denote the mass of the lighter mass as m1 and the mass of the heavier mass as m2.
The initial potential energy of the system is given by the sum of the potential energies of both masses:
Initial potential energy = m1 * g * h1 + m2 * g * h2
Where g is the acceleration due to gravity (9.8 m/s^2), h1 is the initial height of the lighter mass above the ground (1.8 m), and h2 is the height of the pulley above the ground (4.8 m).
When the heavier mass hits the ground, it loses all its potential energy and gains an equal amount of kinetic energy. Therefore, the final kinetic energy of the lighter mass is equal to m2 * g * h2.
Setting the initial potential energy equal to the final kinetic energy:
m1 * g * h1 + m2 * g * h2 = m2 * g * h2
Simplifying the equation:
m1 * g * h1 = m2 * g * h2
Dividing both sides by m1:
g * h1 = m2 * g * h2 / m1
h1 = (m2 * g * h2) / (m1 * g)
Cancelling out the 'g':
h1 = (m2 * h2) / m1
Now, we can substitute the given values into the equation to find h1:
h1 = (m2 * h2) / m1
h1 = (m2 * 4.8) / m1
We can also calculate the time it takes for the heavier mass to hit the ground using the kinematic equation:
h2 = (1/2) * g * t^2
Simplifying the equation for t:
t = sqrt(2 * h2 / g)
Substituting the given values into the equation to find t:
t = sqrt(2 * 4.8 / 9.8)
Now, we can calculate the velocity of the lighter mass at the instance the heavier mass hits the ground using the equation:
v = g * t
Substituting the given value of g and the calculated value of t, we can find the velocity of the lighter mass.
(b) To find the maximum height reached by the lighter object after the system is released, we can equate the initial potential energy of the system to the final potential energy of the lighter mass.
The initial potential energy of the system is the same as in part (a):
Initial potential energy = m1 * g * h1 + m2 * g * h2
The final potential energy of the lighter mass is given by:
Final potential energy = m1 * g * h'
Where h' is the maximum height reached by the lighter mass.
Equating the initial and final potential energies:
m1 * g * h1 + m2 * g * h2 = m1 * g * h'
Simplifying the equation:
h' = h1 + (m2 * h2) / m1
Substituting the given values into the equation, we can find the maximum height reached by the lighter mass.