Finding minimum value of force P given static friction

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In summary, to solve for the minimum value of P that will produce motion when us = 0.400, differentiate your expression for P with respect to Theta and set it to 0, and solve for the root.
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Homework Statement

A crate of weight Fg is pushed by a force P on a horizontal
floor. (a) If the coefficient of static friction is us and P
is directed at angle theta below the horizontal, show that the
minimum value of P that will move the crate is given by

P = ( us*Fg*sec(theta)) / (1 - us*tan(theta))

(b) Find the minimum value of P that can produce motion
when us " 0.400

The Attempt at a Solution

All right, I managed to do (a) on my own, but I'm dead stuck on (b). I've only gotten as far as inputting us = 0.400 in, and all I know is that I need to find an angle such that I get a minimum for P. I'm unsure as to how to solve this problem without inputting a bunch of angles and using trial-and-error to get my answer. Can anyone help me?

Differentiate your expression for P with respect to Theta and set it to 0, find the minimum in the usual way. I think that will work.

You will need to solve that equation. It's not as bad as it looks. Straight away you can simplify it by multiplying top and bottom of the second fraction by 1-0.4tan(x), then the denominators for both are the same, so you can throw them away and you have:

$$\frac{0.16}{cos^{3}(x)}+\frac{0.4sin(x)}{cos^{2}(x)}-\frac{0.16sin^{2}(x)}{cos^{3}(x)}=0$$

You can solve this to get the root you want in a simpler form than it is given on the WolframAlpha page.

Keep in mind that 1-sin^2(x)=cos^2(x), this property is useful in the process of solving the equation.

To find the minimum value of P for a coefficient of static friction of 0.400, we can use the equation from part (a) and set it equal to P. This will give us an equation in terms of only theta, which we can then solve for to find the minimum angle.

P = ( us*Fg*sec(theta)) / (1 - us*tan(theta))

Substituting us = 0.400, we get:

P = (0.400*Fg*sec(theta)) / (1 - 0.400*tan(theta))

To find the minimum value of P, we need to find the minimum value of theta that will give us a positive P. This means we need to find the maximum value of the denominator, since the numerator will always be positive.

Setting the derivative of the denominator equal to zero and solving for theta, we get:

d/dtheta (1 - 0.400*tan(theta)) = 0

-0.400*sec^2(theta) = 0

sec^2(theta) = 0

theta = pi/2

So, the minimum value of P for a coefficient of static friction of 0.400 occurs when theta = pi/2, or when the force P is perpendicular to the floor. Plugging this value into the equation, we get:

P = (0.400*Fg*sec(pi/2)) / (1 - 0.400*tan(pi/2))

P = 0.400*Fg*sec(pi/2)

P = 0.400*Fg*infinity

Since sec(pi/2) is infinite, we can see that the minimum value of P is also infinite. This means that the crate will not move unless the force P is greater than infinity, which is impossible. Therefore, there is no minimum value of P for a coefficient of static friction of 0.400 and the crate will not move.

1. What is the definition of static friction?

Static friction is the force that opposes the motion of an object when it is in a state of rest or stationary on a surface.

2. How is the minimum value of force P calculated?

The minimum value of force P can be calculated by setting the equation for static friction (Fs = μN) equal to the given force P and solving for the coefficient of static friction (μ).

3. What factors affect the minimum value of force P?

The minimum value of force P is affected by the coefficient of static friction, the normal force between the object and the surface, and the angle of the surface.

4. Can the minimum value of force P be negative?

No, the minimum value of force P cannot be negative as it represents the minimum force required to overcome static friction and initiate motion. A negative value would indicate that the object is already in motion.

5. How is the coefficient of static friction determined?

The coefficient of static friction can be determined experimentally by gradually increasing the force applied to the object until it starts to move and recording the corresponding values for force and normal force. The coefficient can then be calculated using the equation μ = Fs/N.

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