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| <title>On the horizontal table is a body of mass m_0. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass m_0 from its place, if the coefficient of friction of the body against the table is \mu? </title> | | <title>On the horizontal table is a body of mass m_0. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass m_0 from its place, if the coefficient of friction of the body against the table is \mu? </title> |
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| <h3 id="back-link"><a href="../../#2.3">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.3">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.3.49.$ On the horizontal table is a body of mass $m_0$. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass $m_0$ from its place, if the coefficient of friction of the body against the table is $\mu$? | | $2.3.49.$ On the horizontal table is a body of mass $m_0$. A non-stretchable thread is attached to the body and thrown over the block. A spring is attached to the free end of the thread. How much weight should the load be attached to the spring so that, when descending, it can move the body of mass $m_0$ from its place, if the coefficient of friction of the body against the table is $\mu$? |
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| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.3.49.png" | | <img src="2.3.49.png" |
| loading="lazy" width="200" /> | | loading="lazy" width="200" /> |
| <figcaption> | | <figcaption> |
| For problem $2.3.49$ | | For problem $2.3.49$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.3.49_1.png" | | <img src="2.3.49_1.png" |
| loading="lazy" width="300" /> | | loading="lazy" width="300" /> |
| <figcaption> | | <figcaption> |
| Forces acting on the system | | Forces acting on the system |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| State of equilibrium for body $m_0$ | | State of equilibrium for body $m_0$ |
| $$N = m_0g$$ | | $$N = m_0g$$ |
| $$F_\text{fr} = \mu N = \mu m_0 g$$ | | $$F_\text{fr} = \mu N = \mu m_0 g$$ |
| Conservation of energy for the attached weight | | Conservation of energy for the attached weight |
| $$\frac{k\Delta x^2}{2} = mg(l+\Delta x) - mgl$$ | | $$\frac{k\Delta x^2}{2} = mg(l+\Delta x) - mgl$$ |
| From where | | From where |
| $$\frac{k\Delta x}{2} = mg$$ | | $$\frac{k\Delta x}{2} = mg$$ |
| Considering | | Considering |
| $$T=k\Delta x \Rightarrow \mu m_0 g = 2mg$$ | | $$T=k\Delta x \Rightarrow \mu m_0 g = 2mg$$ |
| So, we get the desired mass | | So, we get the desired mass |
| $$\boxed{m=\frac{\mu m_0}{2}}$$ | | $$\boxed{m=\frac{\mu m_0}{2}}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$m=\frac{\mu m_0}{2}$$ | | $$m=\frac{\mu m_0}{2}$$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Almaskhan Arsen<br> | | Almaskhan Arsen<br> |
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