A 30.0-kg child sits on one end of a long uniform beam having a mass of 20.0 kg, and a 43.0-kg child sits on the other end. The beam balances when a fulcrum is placed below the beam a distance of 1.10 m from the 30.0-kg child. How long is the beam?
The answer is 1.98, not sure why.
a. 2.12 m
b. 2.07 m
c. 1.98 m
d. 2.20 m
e. 1.93 m
which one ?
sum moments about the fulcrum...
1.1*30-(L-1.1)43 - 20(L/2 -1.1)=0
solve for L
Well, I have to say, that's quite the balancing act! It sounds like a real-life see-saw situation. To find the length of the beam, we can use the principle of torque balance. Since the beam is in equilibrium, the sum of the torques on both sides must be zero.
Now, let me put on my math clown shoes and juggle some numbers for you. The torque on one side of the beam would be the product of the child's weight (30.0 kg) and the distance between the fulcrum and the child (let's call it x). On the other side, it would be the product of the other child's weight (43.0 kg) and the distance between the fulcrum and this child (which would be the total length of the beam minus x).
So, the equation would look like this:
(30.0 kg)(x) = (43.0 kg)(total length of the beam - x)
Now, let's solve for x and find out how far the fulcrum should be from the first child.
30.0x = 43.0(total length of the beam - x)
If we simplify this equation, we get:
30.0x = 43.0(total length of the beam) - 43.0x
Then, let's bring all the x terms to one side:
30.0x + 43.0x = 43.0(total length of the beam)
Combine like terms:
73.0x = 43.0(total length of the beam)
Divide both sides by 73.0:
x = (43.0/73.0)(total length of the beam)
Now, let's replace x with 1.10 m (the given distance from the first child):
1.10 = (43.0/73.0)(total length of the beam)
To solve for the total length of the beam, we can cross-multiply and divide:
(total length of the beam) = (1.10)(73.0)/43.0
And after doing some wacky math calculations, we find that the total length of the beam is approximately 1.87 meters.
So, hang on tight, because that's how long your balancing act should be!
To solve this problem, we can use the principle of torque. Torque is the rotational equivalent of force and is calculated as the product of the force applied and the distance from the pivot point.
Here's how we can find the length of the beam:
1. Determine the torques exerted by each child. Since the beam is balanced, the total torque exerted by one child must be equal and opposite to the total torque exerted by the other child.
2. The torque exerted by a child can be calculated as the product of their weight (mass × acceleration due to gravity) and the distance from the fulcrum to their position.
3. Let's denote the distance from the fulcrum to the 30.0 kg child as x and calculate the torques.
Torque exerted by the 30.0 kg child = (30.0 kg) × (9.81 m/s^2) × x
Torque exerted by the 43.0 kg child = (43.0 kg) × (9.81 m/s^2) × (x + 1.10 m)
4. Since the beam is balanced, the torques must be equal:
(30.0 kg) × (9.81 m/s^2) × x = (43.0 kg) × (9.81 m/s^2) × (x + 1.10 m)
5. Now, we can solve this equation to find the value of x:
(x) = [(43.0 kg) × (9.81 m/s^2) × (x + 1.10 m)] / (30.0 kg × 9.81 m/s^2)
6. Solve the equation to find the value of x:
x = [(43.0 kg) × (9.81 m/s^2) × (x + 1.10 m)] / (30.0 kg × 9.81 m/s^2)
x = (43.0 kg) / (30.0 kg) × (x + 1.10 m)
x = (43.0/30.0) × (x + 1.10)
x = (43/30)x + (43/30)1.10
(7/30)x = (43/30)1.10
x = (43/30)1.10 × (30/7)
x ≈ 1.28 m
7. Finally, the length of the beam can be calculated as:
Length of the beam = x + 1.10 m
Length of the beam = 1.28 m + 1.10 m
Length of the beam ≈ 2.38 m
Therefore, the length of the beam is approximately 2.38 meters.