I am trying to help my son with his math, but I am stuck!
The advertisement for a new shoe promises runners less slip. The coefficient of friction (μ) between concrete and a new material used in the sole of this shoe is 1.5. The force of friction that causes slip is equal to mg sinθ, where m is a runner’s mass and g is the acceleration due to gravity. The force that prevents slip is μmg cosθ.
At the instant of slip, the force that causes slip is equal to the force that prevents it. Write an equation to show this relationship.
Rewrite the equation with trigonometric expressions on one side of the equation, and substitute the coefficient of friction for the new material.
Use a trigonometric identity to rewrite the equation using only the tangent function.
Solve for θ, the angle at which the new shoe will start to slip.
46.3 deg
53.5
56.3 deg
59.2
m g sin T = u m g cos T
sin T = u cos g
tan T = u
if u = 1.5
tan T = 1.5
T = tan^-1 1.5 = 56.3 degrees
To solve this problem, let's take it step by step.
Step 1: Write equations to show the relationship between the forces.
The force of friction that causes slip is mg sinθ.
The force that prevents slip is μmg cosθ.
At the instant of slip, the force that causes slip is equal to the force that prevents it. Therefore, we can write the equation as follows:
mg sinθ = μmg cosθ
Step 2: Rewrite the equation with trigonometric expressions on one side.
Divide both sides of the equation by mg to simplify it:
sinθ = μcosθ
Step 3: Substitute the coefficient of friction for the new material.
The coefficient of friction (μ) for the new material is given as 1.5. Substitute this value into the equation:
sinθ = 1.5cosθ
Step 4: Use a trigonometric identity to rewrite the equation using only the tangent function.
Here, we can use the identity tanθ = sinθ / cosθ to rewrite the equation:
tanθ = sinθ / cosθ = 1.5cosθ / cosθ = 1.5
Step 5: Solve for θ, the angle at which the new shoe will start to slip.
Now we can solve for θ by taking the inverse tangent (arctan) of both sides of the equation:
θ = arctan(1.5)
Using a calculator, you will find that arctan(1.5) ≈ 56.3 degrees.
Therefore, the angle at which the new shoe will start to slip is approximately 56.3 degrees.
So the correct option is 56.3 deg.