A person 1.90 m tall walks on a horizontal icy surface. The coefficient of friction between the surface of the ice and the person's shoe sole is μ = 0.33 and the length of the person's legs are 1/2 of the person's height. Determine the maximum step length the person can make without slipping forward.
Hint: When walking, a person's weight is evenly distributed on each leg. Express the components of the weight in terms of the angle of walking produced by the person's legs and equate the horizontal component of this force to the force of friction acting at the contact between each sole and the horizontal icy surface.
I know that F=ma and that the angle matters, but what would the forces on the leg be.
did you ever figure this out
In order to determine the maximum step length a person can make without slipping forward, we need to consider the forces acting on the person's legs.
Let's break down the forces acting on each leg when the person is walking:
1. Weight (W): The weight of the person acts vertically downward through the center of mass. Since the person's weight is evenly distributed on each leg, the weight can be divided equally between the two legs.
2. Frictional Force (F): The frictional force acts parallel to the surface of the ice and opposes the motion of the person. Since the person is not slipping forward, the maximum step length will occur when the frictional force has reached its maximum value. The equation for frictional force is F = μN, where μ is the coefficient of friction and N is the normal force.
3. Normal Force (N): The normal force acts perpendicular to the surface of the ice and counteracts the weight of the person. It can be divided equally between the two legs.
Now let's consider the components of the weight and how they relate to the angle of walking:
The person's legs are said to be 1/2 of the person's height, so each leg can be assumed to be at an angle of 30 degrees (since the legs form a right triangle with the horizontal surface).
Using trigonometry, the vertical component of the weight on each leg can be calculated using the equation W_vertical = W * sin(angle), and the horizontal component of the weight can be calculated using the equation W_horizontal = W * cos(angle). Since the weight is divided equally between the two legs, each leg will have half of these values.
Now, equating the horizontal component of the weight (W_horizontal/2) to the frictional force (F) acting on each leg, we can solve for the maximum step length.
Let's denote the maximum step length as L.
Therefore, the equation becomes:
W_horizontal/2 = F
(W * cos(30°))/2 = μN
Now we need to express N in terms of the weight W. Since the person's weight is evenly distributed between the two legs, N can be expressed as N = W/2.
Substituting this into the equation, we get:
(W * cos(30°))/2 = μ * (W/2)
Simplifying this equation gives:
cos(30°) = μ/2
With the given coefficient of friction μ = 0.33, we can solve for the maximum step length, L, using the equation:
L = (W * cos(30°))/(2 * μ)
For the final answer, you need to know the weight of the person. Once you have the weight, plug it into the equation along with the values of cos(30°) and μ to calculate the maximum step length without slipping forward.