<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="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 with a 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.">
<meta property="og:title" content="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 with a 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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<meta property="og:description" content="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 with a 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.">
<title>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 with a 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.</title>
$2.1.20.$ A bar of mass $m$ is placed on the horizontal board. The board is slowly tilted. Determine the dependence of the friction force acting on the bar on the angle of inclination of the board $\alpha$. Coefficient of friction $\mu$.
Dependence of the friction force on the angle of the board
</figcaption>
</figure>
</center>
@@ -78,18 +78,18 @@
<p>
1. The friction force in magnitude cannot exceed the value $F_{fr} = \mu N$, where $N$ is the sum of the projections of all forces on the direction perpendicular to the possible movement. When the body is in equilibrium, the friction force is equal to the sum of the projections of the forces on the direction of movement. Thus, in a state of rest
−
$$ F_{fr} = mg \sin\alpha$$
−
2. Otherwise, for $\mu\geqtg \,\alpha$
−
$$ F_{fr} = \mu mg \cos\alpha$$
+
$$ F_{fr} = mg \sin\alpha$$
+
2. Otherwise, for $\mu\geq\tan\alpha$
+
$$ F_{fr} = \mu mg \cos\alpha$$
3. The maximum value of the friction force will occur at the angle $\alpha_0$
$$\boxed{\alpha_0 = \arctan\mu}$$
</p>
</p>
<h4>Answer</h4>
<p>
−
$$F_{fr} = mg \sin\alpha\text{ with }\tan\alpha\leq\mu$$
−
$$F_{fr} = \mu mg \cos\alpha\text{ with }\tan\alpha\geq\mu$$
+
$$F_{fr} = mg \sin\alpha\text{ with }\tan\alpha\leq\mu$$
+
$$F_{fr} = \mu mg \cos\alpha\text{ with }\tan\alpha\geq\mu$$
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="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 with a 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.">
<meta name="description" content="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 with a 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.">
<meta property="og:title" content="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 with a 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.">
<meta property="og:title" content="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 with a 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.">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
<meta property="og:description" content="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 with a 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.">
<meta property="og:description" content="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 with a 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.">
<title>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 with a 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.</title>
<title>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 with a 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.</title>
$2.1.20.$ A bar of mass $m$ is placed on the horizontal board. The board is slowly tilted. Determine the dependence of the friction force acting on the bar on the angle of inclination of the board $\alpha$. Coefficient of friction $\mu$.
$2.1.20.$ A bar of mass $m$ is placed on the horizontal board. The board is slowly tilted. Determine the dependence of the friction force acting on the bar on the angle of inclination of the board $\alpha$. Coefficient of friction $\mu$.
Dependence of the friction force on the angle of the board
Dependence of the friction force on the angle of the board
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
@@ -78,18 +78,18 @@
<p>
<p>
1. The friction force in magnitude cannot exceed the value $F_{fr} = \mu N$, where $N$ is the sum of the projections of all forces on the direction perpendicular to the possible movement. When the body is in equilibrium, the friction force is equal to the sum of the projections of the forces on the direction of movement. Thus, in a state of rest
1. The friction force in magnitude cannot exceed the value $F_{fr} = \mu N$, where $N$ is the sum of the projections of all forces on the direction perpendicular to the possible movement. When the body is in equilibrium, the friction force is equal to the sum of the projections of the forces on the direction of movement. Thus, in a state of rest
$$ F_{fr} = mg \sin\alpha$$
$$ F_{fr} = mg \sin\alpha$$
2. Otherwise, for $\mu\geqtg \,\alpha$
2. Otherwise, for $\mu\geq\tan\alpha$
$$ F_{fr} = \mu mg \cos\alpha$$
$$ F_{fr} = \mu mg \cos\alpha$$
3. The maximum value of the friction force will occur at the angle $\alpha_0$
3. The maximum value of the friction force will occur at the angle $\alpha_0$
$$\boxed{\alpha_0 = \arctan\mu}$$
$$\boxed{\alpha_0 = \arctan\mu}$$
</p>
</p>
</p>
</p>
<h4>Answer</h4>
<h4>Answer</h4>
<p>
<p>
$$F_{fr} = mg \sin\alpha\text{ with }\tan\alpha\leq\mu$$
$$F_{fr} = mg \sin\alpha\text{ with }\tan\alpha\leq\mu$$
$$F_{fr} = \mu mg \cos\alpha\text{ with }\tan\alpha\geq\mu$$
$$F_{fr} = \mu mg \cos\alpha\text{ with }\tan\alpha\geq\mu$$
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
