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| <title>A body located on a horizontal plane is pulled by the thread in the horizontal direction. Draw a graph of the dependence of the friction force acting on the body from the plane on the tension force of the thread. Initially, the body is motionless. Body weight 10 kg, coefficient of friction 0.51.</title> | | <title>A body located on a horizontal plane is pulled by the thread in the horizontal direction. Draw a graph of the dependence of the friction force acting on the body from the plane on the tension force of the thread. Initially, the body is motionless. Body weight 10 kg, coefficient of friction 0.51.</title> |
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| <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.1.19.$ If you press your finger on a ballpoint pen resting on a hard surface while simultaneously tilting it, then as long as the pen forms a small angle perpendicular to the surface, it will obediently follow the finger of the hand. As soon as the handle tilt angle exceeds a certain maximum value of $\alpha_{max}$, it will slip out from under your finger, no matter how hard or weak you press it. Experiment for yourself and estimate the coefficient of friction between the ball of the pen and the surface on which it rests. | | $2.1.19.$ If you press your finger on a ballpoint pen resting on a hard surface while simultaneously tilting it, then as long as the pen forms a small angle perpendicular to the surface, it will obediently follow the finger of the hand. As soon as the handle tilt angle exceeds a certain maximum value of $\alpha_{max}$, it will slip out from under your finger, no matter how hard or weak you press it. Experiment for yourself and estimate the coefficient of friction between the ball of the pen and the surface on which it rests. |
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| <figcaption> | | <figcaption> |
| For problem $2.1.19$ | | For problem $2.1.19$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p> | | <p> |
| Let's do a visual experiment, put a pen on a piece of paper and slowly move it to the side | | Let's do a visual experiment, put a pen on a piece of paper and slowly move it to the side |
| </p> | | </p> |
| <br> | | <br> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.19/demo.gif" | | <img src="https://savchenkosolutions.com/2/2.1.19/demo.gif" |
| loading="lazy" width="350" /> | | loading="lazy" width="350" /> |
| <figcaption> | | <figcaption> |
| Demonstration of a pen falling | | Demonstration of a pen falling |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| At some point, the pen will go in the opposite direction and slip out from under your hand | | At some point, the pen will go in the opposite direction and slip out from under your hand |
| </p> | | </p> |
| <br> | | <br> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.19/draw.png" | | <img src="https://savchenkosolutions.com/2/2.1.19/draw.png" |
| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| The pen is in a critical condition | | The pen is in a critical condition |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| Using a special program, we will determine the angle of the pen at this moment | | Using a special program, we will determine the angle of the pen at this moment |
| </p> | | </p> |
| <br> | | <br> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.19/angle.jpg" | | <img src="https://savchenkosolutions.com/2/2.1.19/angle.jpg" |
| loading="lazy" width="450" /> | | loading="lazy" width="450" /> |
| <figcaption> | | <figcaption> |
| Measuring the critical angle | | Measuring the critical angle |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| It worked | | It worked |
| $$ \boxed{\alpha = 7^{\circ}} $$ | | $$ \boxed{\alpha = 7^{\circ}} $$ |
| Let's display the forces acting on the handle | | Let's display the forces acting on the handle |
| </p> | | </p> |
| <br> | | <br> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.19/draw1.png" | | <img src="https://savchenkosolutions.com/2/2.1.19/draw1.png" |
| loading="lazy" width="180" /> | | loading="lazy" width="180" /> |
| <figcaption> | | <figcaption> |
| Forces acting on the handle | | Forces acting on the handle |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| Since the friction force is the force of static friction | | Since the friction force is the force of static friction |
| $$ F_{тр} = \mu N $$ | | $$ F_{тр} = \mu N $$ |
| Let us write down the equilibrium condition at the critical moment | | Let us write down the equilibrium condition at the critical moment |
| $$ \left\{\begin{matrix} | | $$ \left\{\begin{matrix} |
| \mu N = F \sin \alpha\\ | | \mu N = F \sin \alpha\\ |
| N = F \cos \alpha | | N = F \cos \alpha |
| \end{matrix}\right. $$ | | \end{matrix}\right. $$ |
| Where | | Where |
| $$ \boxed{\mu = \tan \alpha} $$ | | $$ \boxed{\mu = \tan \alpha} $$ |
| We substitute experimental data | | We substitute experimental data |
| $$ \mu = \tan 7^{\circ} = 0.12 $$ | | $$ \mu = \tan 7^{\circ} = 0.12 $$ |
| </p> | | </p> |
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