A man attempts to support a stack of books horizontally
by applying a compressive force of F = 120 N to the ends of the stack
with his hands. If each book has a mass of 0.95 kg, determine the
greatest number of books that can be supported in the stack. The
coefficient of static friction between the man’s hands and a book
is(μs)h = 0.6 and between any two books (μs)b = 0.4.
n=12
To determine the greatest number of books that can be supported in the stack, we need to consider two factors: the compressive force applied by the man and the frictional forces between the books.
1. Calculate the maximum force of static friction between the man's hands and a book:
The maximum force of static friction (Fsf) can be calculated using the formula:
Fsf = μs * N
where μs is the coefficient of static friction and N is the normal force.
The normal force acting on the book is equal to the book's weight (mg), where m is the mass of the book and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Given that the mass of each book is 0.95 kg, the weight of each book is:
Weight = mass * acceleration due to gravity
Weight = 0.95 kg * 9.8 m/s^2
Weight = 9.31 N
Therefore, the maximum force of static friction between the man's hands and a book is:
Fsf = 0.6 * 9.31 N
Fsf = 5.59 N
2. Calculate the force of static friction between two books:
The force of static friction between two books is calculated in the same way as above, but using the coefficient of static friction for books (μs)b = 0.4.
Therefore, the force of static friction between two books is:
Fsf = 0.4 * 9.31 N
Fsf = 3.72 N
3. Determine the maximum number of books that can be supported:
To find the maximum number of books that can be supported, we need to consider that the force of static friction between the man's hands and a book supports the weight of all the books above it.
Let's denote the number of books in the stack as N.
The total force of static friction applied by the man's hands is given by:
Total Fsf = Fsf + (N - 1) * Fsf
The compressive force applied by the man is given as F = 120 N.
Therefore, we have the equation:
F = Total Fsf
120 N = Fsf + (N - 1) * Fsf
120 N = Fsf * N
Substituting the calculated values, we have:
120 N = 5.59 N * N
Solving for N:
N = 120 N / 5.59 N
N ≈ 21
Therefore, the greatest number of books that can be supported in the stack is approximately 21 books.
To determine the greatest number of books that can be supported in the stack, we need to consider the forces acting on the books and calculate the maximum force that can be applied without causing the books to slip.
First, let's analyze the forces acting on the books:
1. The compressive force applied by the man's hands - F = 120 N
2. Normal force - This is the force exerted by the books on the man's hands in the vertical direction. It is equal to the weight of the books.
3. Friction force between the man's hands and the bottom book - This opposes the horizontal force applied by the man's hands.
4. Friction force between each pair of adjacent books - This prevents sliding between the books.
Now, let's break down the force analysis for different parts:
1. Man's hands and the bottom book:
The normal force (N1) exerted by the bottom book on the man's hands is equal to the weight of the bottom book (mg), where m is the mass of the book and g is the acceleration due to gravity.
N1 = m * g = 0.95 kg * 9.8 m/s^2 = 9.31 N
The maximum friction force (F1) that can be exerted by the man's hands without causing the bottom book to slip is given by:
F1 = μs * N1 = 0.6 * 9.31 N = 5.59 N
2. Friction between adjacent books:
The maximum friction force (F2) between any two adjacent books without causing them to slip is given by:
F2 = μs * N2, where N2 is the force exerted by the upper book on the lower book.
N2 = m * g = 0.95 kg * 9.8 m/s^2 = 9.31 N
F2 = μs * N2 = 0.4 * 9.31 N = 3.72 N
To determine the maximum number of books that can be supported, we need to find the total friction force that can be sustained by the stack of books:
Total friction force = (number of book pairs) * F2 + F1
Let's assume there are n books in the stack. The number of book pairs is (n-1). Therefore:
Total friction force = (n-1) * F2 + F1
Now, we can set up the inequality:
(n-1) * F2 + F1 ≤ F
Substituting the given values:
(n-1) * 3.72 N + 5.59 N ≤ 120 N
Simplifying the inequality:
3.72n + 3.72 - 3.72 + 5.59 ≤ 120
3.72n + 5.59 ≤ 120
3.72n ≤ 120 - 5.59
3.72n ≤ 114.41
n ≤ 114.41 / 3.72
n ≤ 30.77
Since we cannot have a fraction of a book, the greatest number of books that can be supported in the stack is 30.
Therefore, the man can support a maximum of 30 books horizontally.