A worker applies a torque to a nut with a wrench 0.485 m long. Because of the cramped space, she must exert a force upward at an angle of 57.0° with respect to a line from the nut through the end of the wrench. If the force she exerts has magnitude 81.5 N, what magnitude torque does she apply to the nut?
N · m
To find the magnitude of the torque applied to the nut, we can use the formula:
Torque = Force * Distance * Sin(θ)
Where:
- Torque is the magnitude of the torque applied to the nut (in N · m)
- Force is the magnitude of the force applied by the worker (in N)
- Distance is the length of the wrench (in meters)
- θ is the angle between the force and the line connecting the nut to the end of the wrench (in degrees)
Given:
Force = 81.5 N
Distance = 0.485 m
θ = 57.0°
First, let's convert θ from degrees to radians, as the sine function requires the angle to be in radians:
θ (in radians) = (θ in degrees) * (π / 180)
θ (in radians) = 57.0° * (π / 180) ≈ 0.9948 radians
Now, we can substitute the values into the formula:
Torque = 81.5 N * 0.485 m * Sin(0.9948 radians)
Calculating Sin(0.9948 radians):
Sin(0.9948 radians) ≈ 0.834
Substituting the values:
Torque = 81.5 N * 0.485 m * 0.834 ≈ 33.29 N · m
So, the worker applies a torque of approximately 33.29 N · m to the nut.