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?

Well, it seems like you're getting a grip on the situation! Let's calculate the torque you can exert without slipping.

First, we need to find the radius of the knob. The diameter is given as 1.2 cm, so the radius would be half of that, which is 0.6 cm (or 0.006 m).

Next, we'll use the formula for torque, which is torque = force × radius × coefficient of static friction. Plugging in the values, we get:

Torque = 0.62 N × 0.006 m × 0.15

Calculating that, we find that you can exert a maximum torque of approximately 0.000558 Nm without letting the knob slip between your fingers.

So, make sure you don't slip up when you're twisting that dial!

To determine the maximum torque you can exert on the knob without having it slip between your fingers, we need to calculate the static friction force.

The torque (τ) exerted on the knob can be calculated using the formula:

τ = r * F * sin(θ)

where:
- τ is the torque,
- r is the radius of the knob (half of the diameter),
- F is the force applied,
- θ is the angle between the applied force and the radius.

In this case, the radius of the knob (r) is half of the diameter, which is 1.2 cm / 2 = 0.6 cm or 0.006 m.

The force applied (F) is given as 0.62 N.

The coefficient of static friction (μ) is given as 0.15.

To calculate the maximum torque, we need to find the angle (θ) that produces the maximum torque without exceeding the static friction force.

μ = F_friction / F_normal

where:
- F_friction is the force of friction between the knob and your fingers,
- F_normal is the normal force pressing the knob against your fingers.

Since the knob is gripped between your thumb and forefinger, the normal force (F_normal) is equal to the force applied (F).

F_friction = μ * F

To find θ, we can use the equation:

sin(θ) = F_friction / F

Now we can plug in the values and calculate:

F_friction = 0.15 * 0.62 = 0.093 N

sin(θ) = 0.093 / 0.62 = 0.15

Taking the inverse sine of both sides:

θ = sin^(-1)(0.15) = 8.68 degrees

Now we can calculate the torque using the maximum angle value:

τ = (0.006 m) * (0.62 N) * sin(8.68 degrees)

τ ≈ 0.00026 Nm

Therefore, the most torque you can exert on the knob without having it slip between your fingers is approximately 0.00026 Nm.

To determine the maximum torque you can exert without having the knob slip, we need to consider the static friction force between the knob and your fingers. Here's how you can calculate it:

1. Find the radius of the knob: The diameter of the knob is given as 1.2 cm, so the radius (r) is half of that: r = 1.2 cm / 2 = 0.6 cm = 0.006 m.

2. Calculate the lever arm: The lever arm is the distance between the axis of rotation (center of the knob) and the point where the force is applied (your fingers). Since you grip the knob with your thumb and forefinger, the lever arm (L) can be estimated as the distance between the center of the knob to the average distance between your thumb and forefinger. Let's assume this distance is approximately 1/3 of the knob's diameter:

Lever Arm = (1/3) * diameter = (1/3) * (1.2 cm) = 0.4 cm = 0.004 m.

3. Calculate the maximum static friction force: The maximum static friction force (F_static_max) can be found by multiplying the coefficient of static friction (μ_static) by the normal force (F_normal). The normal force is equal to the force applied by your grip:

F_static_max = μ_static * F_normal.

Given that the coefficient of static friction (μ_static) is 0.15 and the force applied (F_normal) is 0.62 N:

F_static_max = 0.15 * 0.62 N = 0.093 N.

4. Calculate the maximum torque: The torque (τ) is given by the product of the force applied perpendicular to the lever arm and the lever arm itself:

Torque = F_static_max * L.

Plugging in the values we found:

Torque = 0.093 N * 0.004 m = 0.000372 Nm.

Therefore, the most torque you can exert on the knob without having it slip between your fingers is approximately 0.000372 Nm.

M=F(fr)•(d/2) =μ•F•(d/2)=

0.15•0.62•1.2•10⁻²/2 =0.000558 N•m