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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).

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# Pushing a lawnmower. picture included

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