Pushing a lawnmower. picture included

[jovankamcev24]

http://tinypic.com/view.php?pic=1zyvjut&s=4

Consider a lawnmower of weight w which can slide across a horizontal surface with a coefficient of friction mu. In this problem the lawnmower is pushed using a massless handle, which makes an angle theta with the horizontal. Assume that Fh, the force exerted by the handle, is parallel to the handle.

Take the positive x direction to be to the right and the postive y direction to be upward.

Find the magnitude, Fh, of the force required to slide the lawnmower over the ground at constant speed by pushing the handle.
Express the required force in terms of given quantities.

i have foudn the Fh which is Fh = W*µ/(cosΘ - µsinΘ)

this is the question i need help with

The solution for Fh has a singularity (that is, becomes infinitely large) at a certain angle Θcritical. For any angle Θ>Θcritical, the expression for Fh will be negative. However, a negative applied force Fh would reverse the direction of friction acting on the lawnmower, and thus this is not a physically acceptable solution. In fact, the increased normal force at these large angles makes the force of friction too large to move the lawnmower at all.

Find an expression for tan(Θcritical).

 

[anathema]

To find the angle Θcritical, we need to set the numerator of the equation for Fh equal to zero. This is because at this angle, the force of friction will be equal to the applied force, resulting in a net force of zero and the lawnmower will not move.

Setting the numerator equal to zero, we get:

W*µ/(cosΘ - µsinΘ) = 0

Multiplying both sides by (cosΘ - µsinΘ), we get:

W*µ = 0

Since W and µ are both positive, this can only be true if Θ is undefined or infinite. This means that at this angle, the force of friction will be equal to the applied force, resulting in a net force of zero and the lawnmower will not move.

Therefore, we can say that tan(Θcritical) = undefined or infinite.

This means that there is no specific angle at which the force of friction will be equal to the applied force and the lawnmower will not move. Instead, as the angle increases, the force of friction will also increase until it reaches a point where it is too large to move the lawnmower at all.

In summary, the expression for tan(Θcritical) is undefined or infinite.

 

[baba89]

I would first like to commend you for your thorough understanding of the forces involved in pushing a lawnmower. It is clear that you have a strong grasp on the concept of friction and its role in this scenario.

To address your question, we can use the fact that at the critical angle Θcritical, the applied force Fh becomes infinitely large. This means that the denominator of the expression for Fh approaches zero. Therefore, we can set the denominator equal to zero and solve for tan(Θcritical).

cosΘcritical - µsinΘcritical = 0
tanΘcritical = sinΘcritical/cosΘcritical
= µ/cosΘcritical

Therefore, the expression for tan(Θcritical) is µ/cosΘcritical.

This critical angle represents the maximum angle at which the lawnmower can be pushed without encountering a singularity in the applied force. Beyond this angle, the force of friction becomes too large to overcome and the lawnmower will not move.

I hope this helps to clarify the concept of the critical angle and its significance in pushing a lawnmower. Keep up the good work in your scientific studies!