We use a wrench to turn nuts on bolts because they require less force. Consider a hexagonal nut 1 cm in diameter. We can tighten this nut with one of two wrenches, wrench A with lever arm 10 cm and wrench B with lever arm 20 cm. Both wrenches have a very small mass, so you may neglect their masses in this problem. What is the ratio of the total work it would take to tighten the nut one full turn with wrench A to the total work it would take with wrench B?
The torque required is the same. Work is torque times angle rotated through.
Since both wrenches rotate the nut 2 pi raidans, the work required is the same. The longer wrench will required less force on the handle of the wrench, because of its longer lever arm.
To find the ratio of the total work required to tighten the nut one full turn with wrench A to the total work required with wrench B, we need to consider the definition of work and the mechanical advantage of each wrench.
Work is defined as the product of the force applied to an object and the distance over which the force is applied. In this case, the force applied is the force required to tighten the nut, and the distance is the circumference of the nut (since we need to turn it one full turn).
The amount of force required to tighten the nut depends on the torque needed. Torque is the product of the force applied and the lever arm distance, which is the length of the wrench.
Let's calculate the work for each wrench separately:
For wrench A:
The lever arm distance (length of the wrench) is 10 cm, which is 0.1 meters.
The circumference of the nut, which is the distance over which the force is applied, can be calculated using the formula C = π * d, where d is the diameter of the nut.
In this case, the diameter of the nut is 1 cm, which is 0.01 meters.
Therefore, the circumference is C = π * 0.01 = 0.0314 meters.
Now, we need to calculate the force required to tighten the nut using wrench A. The force is equal to the torque divided by the lever arm distance.
Since we want to tighten the nut with one full turn, the force needed is the same as the torque needed.
The torque required can be calculated using the formula τ = F * r, where F is the force and r is the radius of the nut (half the diameter).
In this case, the radius of the nut is 0.005 meters, and to calculate torque, we need to consider the lever arm distance of wrench A, which is 0.1 meters.
Therefore, the torque is τ = F * 0.1 = F * (0.005 * 2).
For wrench B:
The lever arm distance (length of the wrench) is 20 cm, which is 0.2 meters.
The circumference of the nut is the same, 0.0314 meters, as we need to turn it one full turn.
The force required to tighten the nut using wrench B can be calculated similarly to wrench A, considering the lever arm distance and torque.
Now, we can calculate the total work for each wrench:
For wrench A, the work is given by W = F * C, where F is the force required and C is the circumference of the nut.
Wrench A: W_A = (F * (0.005 * 2)) * 0.0314 = F * 0.0314 meters.
For wrench B, the work is also W = F * C.
Wrench B: W_B = (F * (0.01 * 2)) * 0.0314 = F * 0.0628 meters.
To find the ratio of the total work required for wrench A to that of wrench B, we divide W_A by W_B:
Ratio = W_A / W_B = (F * 0.0314) / (F * 0.0628) = 0.5.
Therefore, the ratio of the total work it would take to tighten the nut one full turn with wrench A to the total work it would take with wrench B is 0.5.