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

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

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).

ou should have found that  

F

h

FhF_h

, the force required to push the lawnmower at constant speed, was

F

h

=

μmg

cos(θ)−μsin(θ)

Fh=μmgcos⁡(θ)−μsin⁡(θ)

.

Note that this expression becomes infinite when the denominator equals zero:

cos(θ)−μsin(θ)=0

cos⁡(θ)−μsin⁡(θ)=0

,

or  

tan(

θ

critical

)=

1

μ

tan⁡(θcritical)=1μ

.

(The phrase "  

F

h

FhF_h

has a singularity at angle  

θ

critical

θcriticaltheta_critical

" means that "  

F

h

FhF_h

goes to infinity at a certain angle  

θ

critical

θcriticaltheta_critical

.")

It's not too hard to understand what this means. Suppose you were pushing straight down on the lawnmower (

θ=90

θ=90

degrees). It obviously wouldn't move. But, according to the equation for  

F

h

FhF_h

, when you plug in  

θ=90

θ=90

degrees, you get a negative force (which doesn't make sense).

The more vertical you push, the harder it gets to move the lawnmower. At  

θ=

θ

critical

θ=θcritical

, it gets impossible to move it. The force required to move it goes to infinity; you have to push infinitely hard.

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Correct. Followup.  

You should have found that , the force required to push the lawnmower at constant speed, was

.

Note that this expression becomes infinite when the denominator equals zero:

,

or  

.

(The phrase " has a singularity at angle " means that " goes to infinity at a certain angle .")

It's not too hard to understand what this means. Suppose you were pushing straight down on the lawnmower ( degrees). It obviously wouldn't move. But, according to the equation for , when you plug in degrees, you get a negative force (which doesn't make sense).

The more vertical you push, the harder it gets to move the lawnmower. At , it gets impossible to move it. The force required to move it goes to infinity; you have to push infinitely hard.

End of followup.

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