I would like to ask about a part of a question I saw in an IG textbook, the question starts as "A smooth bead is threaded on a light inextensible string, the ends of the string are attached to the ceilling, the bead is acted on by a horizontal force F and the bead is in equilibrium….. ", the point is that the answer assumes that the tension on both sides of the string are equal, why is this true?
Static Equilibrium problem
classical-mechanicsmathematical physicsphysics
Related Solutions
Try to convince yourself that the free body diagram below is correct.
- $T \to$tension of the string
- $R_1 \to$Force exerted by the pulley on the string ($4\sqrt2N$ given)
- $R \to$Normal reaction force on $P$ by the table
- $\mu R \to$Static friction working on $P$
Here $\theta=45^{\circ}$
So for the following 3 systems we've got 4 equilibrium:
\begin{align} &\text{Pulley:}&R_1&=2T\cos\theta\tag{i}\\ &\text{Ball Q:}&T&=m_qg\tag{ii}\\ &\text{Ball P:}&T&=\mu R\tag{iii}\\ &\text{Ball P:}&R&=m_pg\tag{iv}\\ \end{align}
From $(i)$ we get : $$4\sqrt2=2T\cos(45^{\circ}) = \frac{2T}{\sqrt2}\\ \therefore T=4N$$ From $(ii)$ we get : $$T=m_qg=10m_q(\mathrm{taking}\ \ g=10m/s^2)\\ \therefore m_q=\frac{T}{10}=\frac{4}{10}=0.4kg$$
Your calculation is incorrect. In this situation the tension shall be $\frac {40}{9} g N$ and the acceleration $\frac{4}{9} g$.
The equations will be -
Total mass * Net acceleration of system = Total downward force
and,
Net downward force - Tension = Mass of B * Acceleration if B
Mathematically,
$$ 18.a = 8.g$$ and,
$$8g - T = 8.a$$.
Subsisting the value of $a$ from the first equation to the second shall give you $T$.
To gain a better understanding for such problems, note that since the the two bodies are connected to form a single system, they must have the same acceleration, $a$. This acceleration shall be the ratio of the total force acting upon and the total mass if the system (see the first equation), as stated by Newton in his second law.
Now, notice that the same reasoning about Newton's law holds individually for the two bodies. (See second equation with regard to body B). It may be helpful to remember that the since the table is smooth, the only force acting on body A, is the tension in the string.
Note that drawing free body diagrams, is in general, a great help in such problems and shall allow you to quickly go through whatever reasoning (like the above) is required by the physical situation.
Best Answer
There are (possibly hidden) assumptions in the problem statement, namely that there is no frictional force between the bead and the string and the part of the string in contact with the bead has zero weight. So although the string may bend around its region of contact with the bead, the tension remains equal throughout that piece of the string; there is no tangential component of any other force (from the bead or from gravity) at any point along that part of the string's path.
If we allow the bead to exert a significant frictional force on the string then we are no longer justified in assuming the tension is equal on both sides. Likewise if the part of the string in contact with the bead has significant weight.
The descriptions of the bead and string as "smooth" and "light" are presumably indicators that we are supposed to assume negligible friction and negligible weight of that portion of the string.