
You press a book perpendicularly against a vertical wall with a force of 27  N
 . The coefficient of static friction is 0.80 between your hand and the book and 0.50 between the book and the wall. What is the maximum book mass you can hold this way and keep from slipping?

To solve this problem, we need to analyze the forces acting on the book and the conditions for static equilibrium.

### Forces Acting on the Book:
1. **Normal force** (\( F_{\text{hand}} \)): This is the force with which you're pressing the book against the wall. Given in the problem, \( F_{\text{hand}} = 27 \, \text{N} \).
2. **Weight of the book** (\( W \)): This is the gravitational force pulling the book down, where \( W = mg \) (with \( m \) being the mass of the book and \( g = 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity).
3. **Frictional forces**: 
   - **Between the book and the wall**: The frictional force \( F_{\text{friction, wall}} = \mu_{\text{wall}} F_{\text{hand}} \), where \( \mu_{\text{wall}} = 0.50 \).
   - **Between your hand and the book**: The frictional force \( F_{\text{friction, hand}} = \mu_{\text{hand}} F_{\text{hand}} \), where \( \mu_{\text{hand}} = 0.80 \).

For the book to remain stationary and not slip, the frictional forces must counteract the weight of the book. Thus, the total frictional force must equal the gravitational force:

\[
F_{\text{friction, total}} = W = mg
\]

The total frictional force comes from the friction between your hand and the book as well as between the book and the wall:

\[
F_{\text{friction, total}} = F_{\text{friction, wall}} + F_{\text{friction, hand}} = \mu_{\text{wall}} F_{\text{hand}} + \mu_{\text{hand}} F_{\text{hand}}
\]

### Now, plug in the values:
\[
F_{\text{friction, total}} = (0.50 \times 27 \, \text{N}) + (0.80 \times 27 \, \text{N}) = 13.5 \, \text{N} + 21.6 \, \text{N} = 35.1 \, \text{N}
\]

This frictional force must equal the weight of the book:

\[
mg = 35.1 \, \text{N}
\]

Now, solve for \( m \):

\[
m = \frac{35.1 \, \text{N}}{9.8 \, \text{m/s}^2} = 3.58 \, \text{kg}
\]

### Conclusion:
The maximum mass of the book you can hold without it slipping is \( 3.58 \, \text{kg} \).

---

Do you need any more details or have any questions?  
Here are some related questions you might find helpful:

1. What is the effect of increasing the normal force applied by the hand?
2. How would the problem change if the coefficient of friction between the wall and the book were lower?
3. What would happen if you reduced the coefficient of friction between your hand and the book?
4. How would you calculate the minimum force needed to hold a heavier book?
5. What if the wall were inclined at an angle? How would that change the result?

**Tip:** When dealing with static friction problems, always ensure the total frictional force can counteract the weight for equilibrium.

You press a book perpendicularly against a vertical wall with a force of 27 N. The coefficient of static friction is 0.80 between your hand and the book and 0.50 between the book and the wall. What is the maximum book mass you can hold this way and keep from slipping?

Determine the maximum mass of a book that can be held against a vertical wall without slipping, given a force of 27 N, with coefficients of static friction 0.80 and 0.50. This problem involves key physics concepts of statics, forces, and friction, suitable for Grades 9-12.

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