## Section 6.18 Forces Bootcamp

### Subsection 6.18.1 Second Law

###### Problem 6.18.1. Finding Mass Ratios From Acceleration Ratios and Inventing Units of Mass.

Follow the link: Checkpoint 6.2.2.

### Subsection 6.18.2 Third Law

###### Problem 6.18.2. Forces on the Car and the Tow Truck.

Follow the link: Checkpoint 6.3.3.

### Subsection 6.18.3 Net Force

###### Problem 6.18.3. Net Force on a Box Pulled by Two Forces in Perpendicular Directions - Example.

Follow the link: Example 6.4.2.

###### Problem 6.18.4. Net Force of Two Forces Simple Case.

Follow the link: Checkpoint 6.4.3.

###### Problem 6.18.5. Net Force of Two Forces General Case.

Follow the link: Checkpoint 6.4.4.

###### Problem 6.18.6. Net Force of Three Forces Simple Case.

Follow the link: Checkpoint 6.4.5.

###### Problem 6.18.7. Three Balanced Forces - Find Magnitude and Direction of One.

Follow the link: Checkpoint 6.4.6.

###### Problem 6.18.8. Four Balanced Forces - Find Magnitudes of Two.

Follow the link: Checkpoint 6.4.7.

### Subsection 6.18.4 Weight

###### Problem 6.18.9. Weight of a Dumb-bell-Shaped Object.

Follow the link: Checkpoint 6.7.3.

### Subsection 6.18.5 Normal Force

###### Problem 6.18.10. A Box on Floor Pushed Against a Wall.

Follow the link: Example 6.8.3.

###### Problem 6.18.11. Normal Force on a Book Against the Wall.

Follow the link: Checkpoint 6.8.7.

### Subsection 6.18.6 Static Friction Force

###### Problem 6.18.12. Determining Static Friction.

Follow the link: Checkpoint 6.9.3.

###### Problem 6.18.13. Maximum Static Friction.

Follow the link: Checkpoint 6.9.7.

###### Problem 6.18.14. Finding Coeficient of Static Friction.

Follow the link: Checkpoint 6.9.9.

### Subsection 6.18.7 Kinetic Friction Force

###### Problem 6.18.15. Kinetic Friction - Accelerating.

Follow the link: Checkpoint 6.10.2.

###### Problem 6.18.16. Kinetic Friction - Constant Velocity.

Follow the link: Checkpoint 6.10.7.

###### Problem 6.18.17. Motion on an Inclined Plane with Kinetic Friction.

Follow the link: Checkpoint 6.10.9.

### Subsection 6.18.8 Drag Force

###### Problem 6.18.18. Computing Viscous Drag Force Given the Speed.

Follow the link: Checkpoint 6.12.4.

###### Problem 6.18.19. Computing Inertial Drag Force.

Follow the link: Checkpoint 6.12.5.

###### Problem 6.18.20. Drag Force and the Terminal Velocity.

Follow the link: Checkpoint 6.12.6.

###### Problem 6.18.21. Comparison of Inertial Drag Forces on Two Airplanes.

Follow the link: Checkpoint 6.12.8.

###### Problem 6.18.22. (Calculus) Speed and Distance on a Skier with Inertial Drag Force.

Follow the link: Checkpoint 6.12.9.

###### Problem 6.18.23. (Calculus) Practice with a Friend: Accounting for Air Drag on a Bob Sled.

Follow the link: Checkpoint 6.12.11.

### Subsection 6.18.9 Spring Force

###### Problem 6.18.24. A Block Hanging from a Spring - The Stretch of Spring.

Follow the link: Checkpoint 6.13.3.

### Subsection 6.18.10 Coupled Motion

###### Problem 6.18.25. Force Between Two Books Moving Together.

Follow the link: Checkpoint 6.15.3

###### Problem 6.18.26. Force Between Two Books Stacked with One Book Pushed from Side.

Follow the link: Checkpoint 6.15.8

###### Problem 6.18.27. A Block Pushed on Another Block and Held in Place by Static Friction.

Follow the link: Checkpoint 6.15.12

###### Problem 6.18.28. Two Blocks Moving Vertically Coupled Through an Ideal Pulley.

Follow the link: Checkpoint 6.15.15

###### Problem 6.18.29. Two Blocks Coupled Through an Ideal Pulley, one Block Moving Horizontally.

Follow the link: Checkpoint 6.15.17

###### Problem 6.18.30. Practice with a Friend: Coupled Motion of Two Blocks on Two Sides of a Fixed Wedge.

Follow the link: Checkpoint 6.15.20

###### Problem 6.18.31. Block on Incline of an Accelerating Wedge.

Follow the link: Checkpoint 6.15.22

###### Problem 6.18.32. Practice with a Friend: Block on Incline of an Accelerating Wedge.

Follow the link: Checkpoint 6.15.25

###### Problem 6.18.33. Block not Sliding on Frictionless Wedge.

Follow the link: Checkpoint 6.15.26

###### Problem 6.18.34. Block not Sliding in Motion of Three Blocks and a Pulley.

Follow the link: Checkpoint 6.15.29

### Subsection 6.18.11 Dynamics of Circular Motion

###### Problem 6.18.35. Tension in a Pendulum.

Follow the link: Checkpoint 6.16.3

###### Problem 6.18.36. Conical Pendulum.

Follow the link: Checkpoint 6.16.6

###### Problem 6.18.37. Car Rounding a Turn on a Slippery Banked Road.

Follow the link: Checkpoint 6.16.9

###### Problem 6.18.38. Practice with a Friend: Rolling a Ball Inside a Funnel.

Follow the link: Checkpoint 6.16.12

###### Problem 6.18.39. Motion in a Vertical Circle.

Follow the link: Checkpoint 6.16.14

###### Problem 6.18.40. Practice with a Friend: How Hard Does the Toy Car Press on the Track?

Follow the link: Checkpoint 6.16.17

###### Problem 6.18.41. A Person at the Wall of a Spinning Drum.

Follow the link: Checkpoint 6.16.18

###### Problem 6.18.42. Practice with a Friend: Circular Motion of a Block Supported by a Hanging Mass.

Follow the link: Checkpoint 6.16.21

### Subsection 6.18.12 Dynamics in Polar Coordinates

###### Problem 6.18.43. (Calculus) Rock Tied to String Moving in a Circle.

Follow the link: Checkpoint 6.17.1

###### Problem 6.18.44. (Calculus) Equation of Motion of a Simple Pendulum.

Follow the link: Checkpoint 6.17.3

###### Problem 6.18.45. (Calculus) Movement of a Bead on a Spinning Frictionless Rod.

Follow the link: Checkpoint 6.17.6

###### Problem 6.18.46. (Calculus) A Block Attached to a String Rotated on a Table with Length of String Changing.

Follow the link: Checkpoint 6.17.9

###### Problem 6.18.47. Practice with a Friend: (Calculus) Length of String Changing at Constant Acceleration.

Follow the link: Checkpoint 6.17.11

###### Problem 6.18.48. Practice with a Friend: (Calculus) Angular Speed and Tension of a Simple Pendulum.

Follow the link: Checkpoint 6.17.12

### Subsection 6.18.13 Miscellaneous

###### Problem 6.18.49. Balancing Spring Force by Static Friction.

A \(30\text{-kg}\) box on a smooth floor is attached to a spring of spring constant \(100\text{ N/m}\) and pulled horizontally by an increasing force \(F \) until the point when the box starts to slide. At that instant, \(F = 75\text{ N}\text{.}\) The coefficient of static friction between the bottom of the box and the floor surface is \(0.2\text{.}\)

At the instant when box is about to slide, by how much is the spring stretched?

Balance forces acting on the block.

\(0.161\text{ m}. \)

We start by identifying forces on the block as in Figure 6.18.51. Note that static friction has its maximum value at the instant of interest.

Since acceleration of the block is zero, forces on the block are balanced. Since every force is along one or the other axis, it is very easy to write equations of motion along the two axes directly. Lets also just use \(F_{s}^{\text{max}} = \mu_s F_N \text{.}\) Also, lets use \(F_{sp} = k \Delta l\text{.}\) We need to find \(\Delta l\text{.}\)

Solve the second equation for \(F_N \) and use that in the first equation, which we solve for \(\Delta l\text{.}\)

Putting the numerical values in this equation we get

###### Problem 6.18.52. An Athlete Pulling on A Rope - Balancing Forces to Find Minimum \(\mu_s\).

A \(100\text{-kg}\) athlete pulls a rope attached to a strong pole at an angle of \(15^{\circ}\) from horizontal such that the tension in the rope devlops to \(800\text{ N}\text{.}\)

(a) Find nomal and frictional forces on the athlete by the floor.

(b) What must be minimum value of the coefficient of static friction between the sole of the shoes of the athlete and the floor so that the athlete does not slip on the floor while he is pulling on the rope?

Draw a Free-Body Diagram of the athlete.

(a) Friction = \(773\text{ N}\text{,}\) Normal = \(774\text{ N}\text{.}\) (b) \(1.0 \text{..}\)

(a) We start by identifying forces on the block in Figure 6.18.54, where I have used symbols for the forces: \(W \) for weight, \(T \) for weight, \(N_L \) for the normal on the left foot, \(N_R \) for the normal on the right foot, \(S_L \) for the static friction on the left foot, and \(S_R \) for the static friction on the right foot.

In the translational motion, we can combine the normal forces into one and call that \(N=N_L+N_R\text{,}\) and similary for the static friction, \(S=S_L+S_R\text{.}\) We will have to keep them separate when we look at rotation.

Since acceleration of the block is zero, the forces are balanced. Since only \(T \) is not along one of the axes, we will need to work out its components. Let \(\theta \) stand for angle \(15^{\circ}\text{.}\)

Now, we can write the equations of motion along the two axes.

Here, the values of \(T \text{,}\) \(\theta\text{,}\) and \(W=mg\text{,}\) are known. Therefore, we solve the first equation to get \(S \text{,}\) and the second one for \(N \text{.}\)

(b)The definition of \(\mu_s \) requires the maximum static friction.

From the problem description, we need the minimum \(\mu_s \text{.}\) That means, if we assume the \(S \) we found was the \(S^{\text{max}}\text{,}\) then the \(\mu_s \) we will get will be the minimum required for the static condition to hold.

###### Problem 6.18.55. Motion of One Pulley and Two Blocks on One Pulley.

Figure 6.18.56 shows a two pulley two block system. While pulley \(\text{P}_1\) only rotates with its center remaining fixed, \(\text{P}_2\) rotates as well as moves. Assume both pulleys to be ideal (massless and frictionless) so that tension on the two sides of each pulley have the same magnitude. Do not assume tensions in the two strings are equal but you can assume that the length of strings do not change during the motion. Find the tensions in the two strings and accelerations of the two blocks.

Use an upward pointed \(y\) axis and write lengths of the two strings in terms of the \(y\) coordinates. From there deduce relation among accelerations of the two blocks.

Partial: \(a_{1y} = \left( \frac{2m_2 - m_1}{m_1 + 4 m_2} \right)\; g \) if positive \(y\) axis pointed up.

Let us use upward pointed \(y\) axis as in Figure 6.18.57 to analyze the motion. Let \(y_1\text{,}\) \(y_p\text{,}\) \(y_2\) denote the positions of the centers of block 1, pulley \(P_2\text{,}\) and block 2 respectively. We will need \(y_p\) to figure out the relation between changes in \(y_1\) and \(y_2\text{.}\)

Let us look at each string to find relations among changes in these \(y\)'s. To facilitate this, let \(y_\text{top}\) be the \(y\) of the fixed pulley. Let us also denote lengths of the strings by \(l_1\) and \(l_2\) respectively. Then, we have

Therefore, we have the following relations among the changes in the coordinates.

From this, we get

This relation between the accelerations and \(y\) equations of motion of the two blocks and the moving pulley (zero mass) will be sufficient to answer the questions about accelerations of the two blocks and the tensions. Using the force diagrams of each block we have the following equations.

Using Eq. (6.18.1), we find

###### Problem 6.18.58. Practice with a Friend: Motion of One Pulley and Three Blocks on One Pulley.

Figure 6.18.59 shows a two pulley three block system. While pulley \(\text{P}_1\) only rotates with its center remaining fixed, \(\text{P}_2\) rotates as well as moves. Assume both pulleys to be ideal (massless and frictionless) so that tension on the two sides of each pulley have the same magnitude. Do not assume tensions in the two strings are equal but you can assume that the length of strings do not change during the motion. Find the tensions in the two strings and accelerations of the three blocks.

No solution provided.