A uniform, 264-N rod that is 2.05 m long carries a 225-N weight at its right end and an unknown weight W toward the left end. When W is placed 47.9 cm from the left end of the rod, the system just balances horizontally when the fulcrum is located 82.7 cm from the right end. Find W. If W is now moved 26.8 cm to the right, how far must the fulcrum be moved to restore balance?
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To solve this problem, we can use the concept of torque. Torque is the rotational equivalent of force and is calculated by multiplying the force applied to an object by the distance from the point of rotation (fulcrum).
First, let's consider the given information. We have a uniform rod that is 2.05 m long, carrying a weight of 225 N at the right end and an unknown weight W toward the left end. The system balances horizontally when the fulcrum is located 82.7 cm from the right end, and W is placed 47.9 cm from the left end.
To find W, we need to set up an equation based on the torques acting on the rod. The torque due to the weight of 225 N at the right end of the rod can be calculated as follows:
Torque1 = Force1 * Distance1
Torque1 = 225 N * (2.05 m - 0.879 m) (converting from cm to m)
Next, we need to calculate the torque due to the unknown weight W placed 47.9 cm from the left end of the rod:
Torque2 = Force2 * Distance2
Torque2 = W * 0.479 m
Since the system balances horizontally, the torque due to the weight at the right end should be equal to the torque due to the weight W at the left end:
Torque1 = Torque2
225 N * (2.05 m - 0.879 m) = W * 0.479 m
Now we can solve for W:
225 N * (2.05 m - 0.879 m) / 0.479 m = W
W ≈ 458.45 N
Therefore, the unknown weight W is approximately 458.45 N.
Now, let's determine how far the fulcrum must be moved to restore balance when W is moved 26.8 cm to the right.
To find the new position of the fulcrum, we can use the principle of moments, which states that the sum of the clockwise torques must equal the sum of the counterclockwise torques.
The total clockwise torque is calculated as follows:
Clockwise Torque = Force1 * Distance1 - W * Distance2
Substituting the given values:
Clockwise Torque = 225 N * (2.05 m - 0.879 m) - 458.45 N * 0.479 m
Now, let's determine the new position of W after it's moved 26.8 cm to the right. The new distance of W from the left end of the rod is:
New Distance2 = 0.479 m + 0.268 m
Next, we can calculate the counterclockwise torque due to the new position of W:
New Counterclockwise Torque = W * Distance2
Substituting the values:
New Counterclockwise Torque = 458.45 N * (0.479 m + 0.268 m)
For the system to balance again, the total clockwise torque must equal the total counterclockwise torque:
Clockwise Torque = New Counterclockwise Torque
225 N * (2.05 m - 0.879 m) - 458.45 N * 0.479 m = 458.45 N * (0.479 m + 0.268 m)
Now, we can solve this equation to find the new position of the fulcrum.
To solve this problem, we can set up equations based on the principle of moments, which states that for an object to be in rotational equilibrium, the sum of the clockwise moments must be equal to the sum of the counterclockwise moments.
Let's start by analyzing the first scenario, where W is placed 47.9 cm from the left end and the fulcrum is located 82.7 cm from the right end.
1. Calculate the moments:
The moment of a force about a point is calculated by multiplying the force by its perpendicular distance from the point.
For the 225-N weight at the right end, the moment is:
Moment1 = (225 N) * (2.05 m - 0.479 m) = 375.225 Nm (clockwise)
For the unknown weight W at 47.9 cm from the left end, the moment is:
Moment2 = W * (0.479 m) (counterclockwise)
2. Set up the equation based on the principle of moments:
According to the principle of moments, the sum of the clockwise and counterclockwise moments must be equal:
Moment1 = Moment2
375.225 Nm = W * 0.479 m
3. Solve for W:
W = 375.225 Nm / 0.479 m
W ≈ 782.86 N
Therefore, W is approximately 782.86 N.
Now, let's analyze the second scenario, where W is moved 26.8 cm to the right. We need to find the distance the fulcrum needs to be moved to restore balance.
4. Calculate the new moments:
For the 225-N weight at the right end, the moment is still:
Moment1 = (225 N) * (2.05 m - 0.479 m) = 375.225 Nm (clockwise)
For W at the new position (47.9 cm + 26.8 cm = 74.7 cm from the left end), the moment is:
Moment3 = W * (0.747 m) (counterclockwise)
5. Set up the new equation based on the principle of moments:
Moment1 = Moment3
375.225 Nm = W * 0.747 m
6. Substitute the value of W we found earlier:
375.225 Nm = 782.86 N * 0.747 m
7. Solve for the new distance of the fulcrum:
Let's call the new distance of the fulcrum x.
375.225 Nm = 782.86 N * (0.747 m + x)
375.225 Nm = 584356.42 Nm + 782.86 N * x
782.86 N * x = 375.225 Nm - 584356.42 Nm
782.86 N * x = -584.731 Nm
x = (-584.731 Nm) / 782.86 N
x ≈ -0.7469 m
Since distance cannot be negative, the fulcrum must be moved approximately 0.747 m (or 74.7 cm) to the left to restore balance.
Therefore, if W is moved 26.8 cm to the right, the fulcrum needs to be moved approximately 0.747 m to the left to restore balance.