How Does Seat Angle Affect Book Sliding in a Car?

[aceaceace]

Homework Statement

A flat book is situated on a seat in a car, which has an angel θ (if θ is zero the seat is horizontal and if it is 90 degrees it is vertical). The coefficient of static friction is μ_s.

(1) What is the largest forward acceleration the car can have, without the book sliding?

(2) What is the largest acceleration the car can have when breaking, without the book sliding?

The Attempt at a Solution

(1) I set up the following equation, assuming the book can only slide backwards.

μ_s*( cos(θ)*m*g - sin(θ)*m*a ) = sin(θ)*m*g + cos(θ)*m*a

Isolating a here gives the right result I think, but after I attemted to solve (2) I thought about, what if the book was sliding the other way, would the equiation be

0 = sin(θ)*m*g + cos(θ)*m*a + μ_s*( cos(θ)*m*g - sin(θ)*m*a )

Isolating a here gives an expression that doesn't really make sense to me.

(2) I set up the following equation, assuming the book can only slide forward.

μ_s*( cos(θ)*m*g + sin(θ)*m*a ) = cos(θ)*m*a - sin(θ)*m*g

Isolating a here gives a similar result that I can't figure out. At certain angles the acceleration becomes infinite.

Am I making a fundamental error somewhere, or am I just failing to see the logic in the results I am getting?

 

[anathema]

Thank you for your post. It seems like you have set up the equations correctly for both scenarios. However, there are a few things to consider:

1. In the first scenario, the book can only slide backwards if the car is accelerating forwards, so the equation should be μs*(cos(θ)*m*g + sin(θ)*m*a) = sin(θ)*m*g - cos(θ)*m*a. This is because the friction force acts in the opposite direction of the car's acceleration.

2. In the second scenario, the book can only slide forwards if the car is decelerating, so the equation should be μs*(cos(θ)*m*g - sin(θ)*m*a) = cos(θ)*m*a - sin(θ)*m*g. This is because the friction force acts in the same direction as the car's deceleration.

3. The equations you have set up give the maximum acceleration that the car can have without the book sliding. This means that if the acceleration of the car exceeds this value, the book will start sliding. So, it is possible for the acceleration to be infinite if the car is at a certain angle and the coefficient of friction is low enough.

I hope this helps clarify your results. Let me know if you have any further questions. Keep up the good work!

 

[thomate1]

It seems like you are on the right track with your equations, but there may be some errors in your calculations or assumptions. Let's break down each scenario separately:

(1) The largest forward acceleration the car can have without the book sliding can be calculated using the following equation:

μs*(m*g*cos(θ) - m*a*sin(θ)) = m*g*sin(θ) + m*a*cos(θ)

Solving for a, we get:

a = (μs*m*g*cos(θ) - m*g*sin(θ)) / (m + μs*sin(θ)*cos(θ))

This equation gives the maximum acceleration without sliding, assuming the book can only slide backwards. If the book is sliding forwards, then the acceleration would be negative and the book would slide backwards. So, the equation you have for this scenario is correct.

(2) The largest acceleration the car can have when braking, without the book sliding, can be calculated using the following equation:

μs*(m*g*cos(θ) + m*a*sin(θ)) = m*a*cos(θ) - m*g*sin(θ)

Solving for a, we get:

a = (μs*m*g*cos(θ) - m*g*sin(θ)) / (m - μs*sin(θ)*cos(θ))

This equation gives the maximum acceleration without sliding, assuming the book can only slide forwards. If the book is sliding backwards, then the acceleration would be negative and the book would slide forwards. So, the equation you have for this scenario is also correct.

In both cases, the acceleration becomes infinite when θ = 90 degrees, which makes sense because at this angle, the book is in a vertical position and there is no friction force to prevent it from sliding.

Overall, your equations and approach seem to be correct. Just make sure to double check your calculations and assumptions to avoid any errors. Great job on thinking through this problem!