A vaulter holds a 20.7 N pole in equilibrium by exerting an upward force, U, with his leading hand, and a downward force, D, with his trailing hand. Let L1 = 0.71 m, L2 = 1.5 m, and L3 = 2.21 m. What are the magnitudes of U and D?
I assume that the pole is horizontal as the vaulter runs up to the bar. I cannot do the problem for you without some idea of the location of the quoted distances in relation to his (or her) hands and the length of the pole.
In any case, set the total moment about either hand equal to zero, and assume the weight of the pole acts through the middle. The will let you solve for the other force.
To find the magnitudes of U and D, we need to use the principle of torque equilibrium. Torque is defined as the force applied multiplied by the perpendicular distance from the pivot point (in this case, where the pole rests against the vaulter's shoulder).
Let's analyze the forces and torques involved:
1. The upward force U exerted by the leading hand creates a clockwise torque.
- Torque of U = U * L1 (clockwise)
2. The weight of the pole (20.7 N) creates a counterclockwise torque.
- Torque of weight = weight * L2 (counterclockwise)
3. The downward force D exerted by the trailing hand creates a clockwise torque.
- Torque of D = D * L3 (clockwise)
Since the pole is in equilibrium (not rotating), the total torque must be zero.
Mathematically, we can express the equilibrium equation as:
clockwise torques = counterclockwise torques
U * L1 + D * L3 = weight * L2
Substituting the given values:
U * 0.71 m + D * 2.21 m = 20.7 N * 1.5 m
Now we have one equation with two unknowns (U and D). However, we can manipulate the equation to solve for one variable in terms of the other.
Rearranging the equation:
U * 0.71 m = 20.7 N * 1.5 m - D * 2.21 m
Simplifying:
U = (20.7 N * 1.5 m - D * 2.21 m) / 0.71 m
Now, we can substitute this expression for U into the equilibrium equation to solve for D.
U * 0.71 m + D * 2.21 m = 20.7 N * 1.5 m
[(20.7 N * 1.5 m - D * 2.21 m) / 0.71 m] * 0.71 m + D * 2.21 m = 20.7 N * 1.5 m
Cancelling out the units:
20.7 N * 1.5 - D * 2.21 + D * 2.21 = 20.7 N * 1.5
Expanding and simplifying:
31.05 N - 2.21 D + 2.21 D = 31.05 N
The D terms cancel out, leaving us with:
31.05 N = 31.05 N
This equation holds true, which means the system is solvable.
Finally, we substitute the value of D into one of the original equations to find the value of U. Once we have both U and D, we can calculate their magnitudes (absolute values).
Since D does not affect the solution, we can choose an arbitrary value for it. Let's say D = 10 N for simplicity:
U * 0.71 m + 10 N * 2.21 m = 20.7 N * 1.5 m
Simplifying:
U * 0.71 m + 22.1 N * m = 31.05 N * m
Subtracting 22.1 N * m from both sides:
U * 0.71 m = 31.05 N * m - 22.1 N * m
Combining like terms:
U * 0.71 m = 8.95 N * m
Dividing both sides by 0.71 m:
U = 8.95 N * m / 0.71 m
Simplifying:
U = 12.61 N
Now we have the magnitude of U. To find the magnitude of D, we use the fact that the sum of the vertical forces must equal the weight of the pole:
U + D = 20.7 N
Substituting the value of U we just found:
12.61 N + D = 20.7 N
Subtracting 12.61 N from both sides:
D = 20.7 N - 12.61 N
Simplifying:
D = 8.09 N
So, the magnitudes of U and D are approximately 12.61 N and 8.09 N, respectively.