A mechanic applies a force of 42 Newtons straight down to a ratchet that is 0.59 meters long. What is the magnitude of the torque when the handle makes a 38 degree angle above the horizontal?
To calculate the magnitude of the torque, we need to use the formula:
Torque = Force * lever arm
The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force. In this case, it is the length of the ratchet, which is 0.59 meters.
However, since the angle is given with respect to the horizontal, we need to find the vertical component of the force applied.
Vertical Force = Force * sin(angle)
Substituting the given values:
Vertical Force = 42 N * sin(38°)
Now we can calculate the torque:
Torque = Vertical Force * Lever Arm
Torque = (42 N * sin(38°)) * 0.59 m
Using a calculator, we find:
Torque ≈ 22.48 N·m (rounded to two decimal places)
Therefore, the magnitude of the torque is approximately 22.48 N·m.
To find the magnitude of the torque, we need to first understand the concept of torque and how to calculate it.
Torque is the measure of the tendency of a force to rotate an object about an axis. It is calculated as the product of the force and the perpendicular distance from the axis of rotation to the line of action of the force.
The formula for torque is:
Torque = Force x Perpendicular Distance
In this case, the force applied by the mechanic is 42 Newtons. The length of the ratchet is given as 0.59 meters.
Now, to calculate the perpendicular distance, we need to consider the given angle. The handle makes a 38 degree angle above the horizontal, which means the remaining angle between the handle and the vertical axis is 90 - 38 = 52 degrees.
Using trigonometry, we can find the perpendicular distance. The perpendicular distance is the length of the ratchet multiplied by the sine of the angle.
Perpendicular Distance = Length of Ratchet x sin(Angle)
Plugging in the given values:
Perpendicular Distance = 0.59 m x sin(52°)
Now, let's calculate the perpendicular distance:
Perpendicular Distance = 0.59 m x 0.788
Perpendicular Distance ≈ 0.465 m
Finally, we can calculate the magnitude of the torque:
Torque = Force x Perpendicular Distance
Torque = 42 N x 0.465 m
Torque ≈ 19.53 Nm
Therefore, the magnitude of the torque when the handle makes a 38 degree angle above the horizontal is approximately 19.53 Nm.