A boy and girl sit on a seesaw of length 4m, balanced at its center. The girl sits at the far end and has a mass of 50kg. The boy is twice as heavy as the girl, and therefore sits midway between his end and the center. Then the girl is given a bag of oranges weighing 5kg, and the seesaw rotates out of balance. When the boy is given a bag of apples, balance is still not restored. He has to place the apples 0.5m behind him for them to be balanced again. What is the weight of the bag of apples?
Let x = mass of Bag of apples {weight of Bag of apples = xg}
Static Equilibrium about Pivot
¦²CCW Torque = ¦²CW Torque: {CW & CCW directions depend on the diagram}
Torque of Girl + bag of oranges {at Right side of pivot} = {50 + 5)(9.8)(2) = 1078 m-N {CW}
Torque of Boy {Left side of pivot} = 100(9.8)(1) = 980 m-N {CCW}
Torque of Bag of apples {Left side of pivot} = x(9.8)(1.5) = 14.7x m-N {CCW}
980 + 14.7x = 1078
14.7x = 98
x = 6.67 kg
ANS = xg = 6.67(9.8) ¡Ö 65 N
Let's solve this step-by-step:
Step 1: Calculate the position of the center of mass of the original system.
Since the seesaw is balanced, the center of mass of the system (CM_system) is at the center of the seesaw, which is halfway between the girl and the boy. Therefore, CM_system = 2m from the girl's end.
Step 2: Calculate the position of the girl's center of mass before adding the bag of oranges.
The halfway point between the girl's end and the center of the seesaw is 2m/2 = 1m from the girl's end. Therefore, the girl's center of mass (CM_girl) is 1m from the girl's end.
Step 3: Calculate the new position of the center of mass of the system after adding the bag of oranges.
Since the girl is 1m from the girl's end and the bag of oranges weighs 5kg, the new center of mass of the system (CM_system_new) is given by the equation:
CM_system_new = (CM_girl * girl_mass + CM_oranges * oranges_mass) / (girl_mass + oranges_mass)
Substitute the values: CM_system_new = (1m * 50kg + CM_oranges * 5kg) / (50kg + 5kg)
Step 4: Calculate the position of the boy's center of mass before adding the bag of apples.
The halfway point between the boy's end and the center of the seesaw is 2m/2 = 1m from the boy's end. Therefore, the boy's center of mass (CM_boy) is 1m from the boy's end.
Step 5: Calculate the position of the boy's center of mass after adding the bag of apples.
Since the boy is 1m from the boy's end and the apples are placed 0.5m behind him, the new center of mass of the boy-apple system (CM_boy-apples) is given by the equation:
CM_boy-apples = (CM_boy * boy_mass + CM_apples * apples_mass) / (boy_mass + apples_mass)
Since CM_boy-apples = CM_system_new, we can substitute the values and solve for CM_apples:
CM_system_new = (1m * 100kg + CM_apples * apples_mass) / (100kg + apples_mass)
Step 6: Set CM_system_new = CM_boy-apples and solve for the weight of the apples (apples_mass).
Using the equation from step 5:
(1m * 100kg + CM_apples * apples_mass) / (100kg + apples_mass) = (1m * 50kg + CM_oranges * 5kg) / (50kg + 5kg)
(100kg + CM_apples * apples_mass) / (100kg + apples_mass) = (50kg + CM_oranges * 5kg) / 55kg
Step 7: Solve the equation from step 6 to find the weight of the apples (apples_mass).
Cross-multiply and simplify:
(100kg + CM_apples * apples_mass) * 55kg = (50kg + CM_oranges * 5kg) * (100kg + apples_mass)
5500kg + CM_apples * apples_mass * 55kg = 5000kg + CM_oranges * 5kg * 100kg + CM_oranges * apples_mass * 5kg
5500kg + 55kg * CM_apples * apples_mass = 5000kg + 500kg * CM_oranges + CM_oranges * apples_mass * 5kg
Step 8: Substitute the known values of CM_apples, apples_mass, CM_oranges, and oranges_mass into the equation from step 7.
Since CM_apples = 1m, CM_oranges = 1m, apples_mass = unknown (let's call it A), and oranges_mass = 5kg, we can substitute these values into the equation:
5500kg + 55kg * 1m * A = 5000kg + 500kg * 1m + 5kg * 1m * A
Step 9: Solve the equation from step 8 to find the weight of the apples (A).
Simplify and solve for A:
5500kg + 55kg * A = 5000kg + 500kg + 5kg * A
5500kg + 55kg * A = 5500kg + 5kg * A
55kg * A - 5kg * A = 5500kg - 5500kg
50kg * A = 0kg
A = 0kg
Step 10: Conclusion
The weight of the bag of apples is 0kg, meaning that the bag is empty.
To solve this problem, we need to determine the weight of the bag of apples.
Let's break down the information given:
1. The seesaw has a length of 4m and is initially balanced at its center.
2. The girl sits at the far end of the seesaw and has a mass of 50kg.
3. The boy is twice as heavy as the girl, so his mass is 2 * 50kg = 100kg.
4. When the girl is given a bag of oranges weighing 5kg, the seesaw rotates out of balance.
5. When the boy is given a bag of apples, the seesaw is still not balanced, and he has to place the apples 0.5m behind him to restore balance.
Let's calculate:
The total mass on each side of the seesaw should be equal for it to be balanced.
Initial balance:
The girl's mass + the boy's mass = 50kg + 100kg = 150kg
When the girl is given a bag of oranges, the seesaw becomes unbalanced. Let's calculate the torque about the center of the seesaw:
Torque of the girl = Distance to center * Mass of the girl
Torque of the boy = Distance to center * Mass of the boy
Since the seesaw is initially balanced, the torques are equal:
Distance to center * Mass of the girl = Distance to center * Mass of the boy
Distance to center * 50kg = Distance to center * 100kg
The distances to the center of the seesaw are not given, but we do know the girl sits at the far end (2m from the center) and the boy sits midway between his end and the center (1m from the center).
(2m) * 50kg = (1m) * 100kg
100kgm = 100kgm
So, the distances to the center of the seesaw are equal.
Now, let's consider the situation when the boy is given a bag of apples, and he has to place them 0.5m behind him to restore balance.
Torque of the girl + Torque of the boy + Torque of the bag of apples = 0
Since the girl's torque (50kg * distance to the center) is counteracted by the torque of the boy (100kg * distance to the center), we have:
Torque of the bag of apples = - (Torque of the girl + Torque of the boy)
Torque of the bag of apples = - (Distance to center * Mass of the girl + Distance to center * Mass of the boy)
Given that the distance from the boy's position to the center is 1m and the distance from the apples to the boy is 0.5m, we can calculate the torque of the bag of apples:
Torque of the bag of apples = - (1m * 50kg + 0.5m * Mass of the apples)
Since the torque is equal to zero (balanced):
0 = - (1m * 50kg + 0.5m * Mass of the apples)
Let's solve this equation to find the mass of the apples.
0 = - (50kg + 0.5m * Mass of the apples)
Rearranging the equation:
50kg = 0.5m * Mass of the apples
Mass of the apples = 50kg / 0.5m
Mass of the apples = 100kg/m
So, the weight of the bag of apples is 100kg.