A combination lock has a 1.2 cm -diameter knob that is part of the dial you turn to unlock the lock. To turn that knob, you grip it between your thumb and forefinger with a force of 0.62 N as you twist your wrist. Suppose the coefficient of static friction between the knob and your fingers is only 0.15 because some oil accidentally got onto the knob.
What is the most torque you can exert on the knob without having it slip between your fingers?
To determine the most torque you can exert on the knob without having it slip between your fingers, you need to consider the friction force and the torque equation.
The torque (τ) exerted on an object is given by the equation τ = r * F * sin(θ), where:
- r is the lever arm (the radius from the rotation axis to the point where the force is applied),
- F is the force applied,
- sin(θ) accounts for the direction of the force relative to the lever arm.
In this case, the torque required to make the knob slip is the maximum torque you can exert without slipping.
To calculate the maximum torque, we need to determine the moment arm (lever arm) and the force applied between the knob and your fingers.
1. Calculate the moment arm (lever arm):
The moment arm is the distance from the center of the knob to the point where the force is applied. Given that the diameter of the knob is 1.2 cm, the radius (r) would be half of that, which is 0.6 cm or 0.006 m.
2. Calculate the force applied:
The force applied while gripping the knob is given as 0.62 N.
3. Calculate the coefficient of static friction:
The coefficient of static friction between the knob and your fingers is given as 0.15.
Now, we can calculate the maximum torque:
τ = r * F * sin(θ)
Since the force is applied perpendicular to the lever arm in this case, sin(θ) is equal to 1.
τ = 0.006 m * 0.62 N * 1
τ = 0.00372 Nm
Therefore, the most torque you can exert on the knob without having it slip between your fingers is 0.00372 Nm.