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?

Repeated post. Same answer.

To find the magnitudes of the upward force (U) and the downward force (D) exerted by the vaulter's hands, we need to use the principle of torque equilibrium. Torque is the rotational equivalent of force and, in equilibrium, the total torque acting on an object is zero.

We can start by calculating the torques exerted by the forces U and D around a chosen pivot point. Let's choose the pivot point as the point where the pole is being held.

The torque exerted by a force is given by the formula: Torque = Force * Lever Arm

For the upward force U, the torque exerted is U * L1 (since the lever arm is L1).

For the downward force D, the torque exerted is -D * L3 (since the lever arm is in the opposite direction to the force).

Since the pole is in equilibrium, the sum of the torques must be zero:

U * L1 - D * L3 = 0

Now, we can use this equation along with the given information:

L1 = 0.71 m
L3 = 2.21 m

Substituting these values into the equation, we get:

U * 0.71 - D * 2.21 = 0

Next, we use another piece of information provided in the question: the vaulter holds the pole in equilibrium with a total force of 20.7 N. This means that the magnitudes of U and D must add up to 20.7 N.

Mathematically, this can be expressed as:

U + D = 20.7

We now have two equations:

U * 0.71 - D * 2.21 = 0 (Equation 1)
U + D = 20.7 (Equation 2)

We can solve these equations simultaneously using any suitable method, such as substitution or elimination. Let's use the method of substitution.

From Equation 2, we can express U in terms of D:

U = 20.7 - D

Substituting this into Equation 1:

(20.7 - D) * 0.71 - D * 2.21 = 0

Distributing and simplifying:

14.697 - 0.71D - 2.21D = 0

Combining like terms:

14.697 - 2.92D = 0

Subtracting 14.697 from both sides:

-2.92D = -14.697

Dividing by -2.92:

D = 5.034

Now that we have the value of D, we can substitute it back into Equation 2 to find U:

U + 5.034 = 20.7

Subtracting 5.034 from both sides:

U = 20.7 - 5.034

U = 15.666

Therefore, the magnitudes of the upward force (U) and the downward force (D) are approximately 15.666 N and 5.034 N, respectively.