Two objects are connected by a light string passing over a light, frictionless pulley as in the diagram. m1 = 4.50 kg, m2 = 2.10 kg, and h = 4.30 m. The 4.50 kg object is released from rest at a point 4.30 m above the floor. Determine the speed of each object when the two pass each other. Determine the speed of each object at the moment the 4.50 kg object hits the floor. How much higher does the 2.10 kg object travel after the 4.50 kg object hits the floor?
To solve this problem, we can use the principles of conservation of energy and Newton's laws of motion.
1. Determine the potential energy of the 4.50 kg object when it is released from rest at a height of 4.30 m above the floor:
Potential energy (PE) = mass (m1) * gravitational acceleration (g) * height (h)
PE = 4.50 kg * 9.8 m/s^2 * 4.30 m
2. Since the system is frictionless, the potential energy of the 4.50 kg object is converted entirely into the kinetic energy of the system as it falls. Thus, we can equate the potential energy to the kinetic energy of the system at that point:
PE = m1gh = (1/2) * m1 * v1^2 + (1/2) * m2 * v2^2
3. Solving for the velocity of object 1 (v1):
v1 = sqrt(2 * g * h)
4. Now, we can use the relationship between the velocities of objects connected by the same string:
v2 = -v1
5. Substituting the value of v1 into the equation for v2:
v2 = -sqrt(2 * g * h)
6. To determine the velocities when the objects pass each other, we need to find the distance traveled by object 2 at that moment. Since both objects have the same speed but opposite directions, initially:
distance = speed * time
7. The time it takes for object 1 to reach the floor can be found using the equation for free fall:
h = 0.5 * g * t^2
Solving for t:
t = sqrt(2h / g)
8. Substitute the value of t into the equation for distance:
distance = v2 * t
9. To determine the velocities when the 4.50 kg object hits the floor, we need to find the time it takes for it to reach the floor. Using the equation for free fall:
h = 0.5 * g * t2^2
Solving for t2:
t2 = sqrt(2h / g)
10. The velocity of the 4.50 kg object when it hits the floor is simply the final velocity after free fall:
v1_f = g * t2
11. To determine the velocity of the 2.10 kg object at the same moment, we can use the relationship between their velocities and masses:
v2_f = -(m1 / m2) * v1_f
12. To find how much higher the 2.10 kg object travels after the 4.50 kg object hits the floor, we need to calculate the difference in heights:
final_height = initial_height - (v1_f^2 / (2 * g))
Following these steps, you should be able to find the answers to the problem.