What is the Optimal Angle for Pulling a Sled to Minimize Force?

[slugglee]

Homework Statement

A Small boy is sledding on a snowy slope and looks very weary as he drags his sled up again after each run. Each time he pulls the sled up he makes an angle of 30 degrees with respect to the ground. The mu of kinetic friction is 0.10. At what angle should the boy be pulling to exert minimum force?

Homework Equations

The Attempt at a Solution

By way of plugging and chugging various numbers, I believe that the angle should be 0. But to get a difinitive answer,our teacher hinted that I would need to use calculus, but I'm not readily famailar with exactly how to apply a sum of all forces equation in that manner.

 

[LowlyPion]

Welcome to PF.

I would suggest that you first draw a diagram of the sled/boy/slope.

Consider that the slope is at angle φ and let the boy be pulling (relative to the slope) at angle θ, such that the normal force on the sled would be given by m*g*cosφ (less the θ component of lift from the boy) and the downward force of gravity would be given by m*g*sinφ less the cosine force of the pull from the boy. .

From that you should be able to create an equation that identifies the Work done to carry himself up the hill, and the work needed to pull the sled up the hill against friction. (Don't forget he is adding to his weight and hence his work in getting up the hill as he pulls at angle θ.)

Ultimately you will want to identify, for any particular φ the minimum of the function of θ in the expression of work.

 

[thomate1]

I would first clarify the problem and gather all the necessary information. The given information includes the angle at which the boy is pulling the sled, the coefficient of kinetic friction, and the desired outcome of finding the angle at which the minimum force is exerted.

Next, I would use the laws of motion and forces to analyze the situation. The boy is pulling the sled at an angle of 30 degrees with respect to the ground, which means that the force he is exerting is not directly opposite to the direction of motion. This results in a component of force that is perpendicular to the direction of motion, which is not helpful in pulling the sled. To minimize this component of force, the angle at which the boy pulls should be zero, meaning that he should pull the sled in a straight line parallel to the ground.

To calculate the minimum force required, I would use the equation F=μmg, where μ is the coefficient of kinetic friction, m is the mass of the sled, and g is the acceleration due to gravity. By plugging in the given values, I can find the minimum force required to pull the sled up the slope.

In order to find the angle at which this minimum force is exerted, I would use calculus. The sum of all forces acting on the sled can be represented by the equation ΣF=ma, where a is the acceleration of the sled. By using trigonometry and differentiating the equation with respect to the angle, I can find the angle at which the force required is minimized.

In conclusion, the angle at which the boy should pull the sled to exert minimum force is 0 degrees, and this can be calculated using the laws of motion and calculus.