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 have no idea how to do this prob.
To solve this problem, you need to consider the forces acting on the person while they are walking. The main forces involved are the weight of the person and the frictional force between the shoe sole and the icy surface. By analyzing these forces, you can determine the maximum step length without slipping forward.
Let's break down the steps to solve this problem:
1. Draw a free-body diagram: Draw a diagram that shows the forces acting on the person standing on the icy surface. Label the weight of the person as W and the force of friction as f.
2. Resolve the weight into components: Since the person's weight is evenly distributed on each leg, we can consider only one leg. We need to resolve the weight into two components: one perpendicular (normal force) to the surface and one parallel (horizontal component) to the surface. We can use trigonometry to do this.
3. Equate the horizontal component of the weight to the force of friction: Since the person is not slipping forward, the maximum step length occurs when the horizontal component of the weight is equal to the force of friction.
4. Calculate the maximum step length: The horizontal component of the weight can be expressed as W * sin(θ), where θ is the angle formed by the leg with the surface. The force of friction can be calculated as μ * (W * cos(θ)). Equate these two forces and solve for θ. Once you have θ, you can use it to calculate the maximum step length, which is given by 2 * (leg length) * sin(θ).
5. Substitute the given values and solve: Plug in the known values (e.g., the person's height, coefficient of friction) to calculate the maximum step length.
By following these steps, you should be able to determine the maximum step length the person can make without slipping forward on the icy surface.