When two or more springs are connected in parallel, we can replace (by removing) all these springs with an equivalent spring (effective spring) whose net effect is same as if all the springs are in parallel connection.
k_p = k₁ + k₂
If n springs are connected in parallel then k_p = nk.

\(\text {Temperature } \quad \mathrm{T}=80+273=353 \mathrm{~K} \)
\(\text {Pressure } P=5 \times 10^{-10} \mathrm{~N} / \mathrm{m}^{2} \)
\(\mathrm{k}_{\mathrm{B}}=\frac{R}{N} \quad \therefore R=K_{B}-N \)
\(P=n R T \)
\(=\mathrm{nk}_{\mathrm{B}} \mathrm{NT} \)
\(\text { Number of molecules } \mathrm{N}=\frac{P}{k_{B} T} \)
\(=\frac{5 \times 10^{-10}}{353 \times 1.3 .8 \times 10^{-23}} \)
\(=\frac{5 \times 10^{-10+23}}{487} \)
\(=0.0102 \times 10^{13} \)
\(\text {Number of molecules }=1.02 \times 10^{11}\)

The factors affecting the surface tension of a liquid are
(i) The presence of any contamination or impurities.
(ii) The presence of dissolved substances
(iii) Electrification and
(iv) Temperature

It is defined as the product of the magnitudes of both the vectors and the cosine of the angle between them.
If there are two vectors \(\overrightarrow { A } \) and \(\overrightarrow { B } \) having an angle θ between then,  \(\overrightarrow { A } \).\(\overrightarrow { B } \)= ABCDθ.
Here, A and B are magnitude of \(\overrightarrow { A } \) and \(\overrightarrow { B } \).
1. It is commutative, i.e., \(\vec{A} \cdot \vec{B}=\vec{B} \cdot \vec{A}\).
2. It obeys distributive law, i.e, \(\vec{A} \cdot(\vec{B}+\vec{C})=\vec{A} \cdot \vec{B}+\vec{A} \cdot \vec{C}\).
3. \((\vec{A} \cdot \vec{B})_{\max }=\mathrm{AB}\), when \(\theta=0^{\circ}\), i.e., when the vectors are parallel.
4. \((\vec{A} \cdot \vec{B})_{\min }=-A B\), when \(\theta=180^{\circ}\), i.e., when the vectors are antiparallel.
5. \(\vec{A} \cdot \vec{B}=0\) when \(\theta=90^{\circ}\), i.e., when the vectors are mutually orthogonal.
6. The self dot product of unit vectors \(\vec{i}, \vec{j} \ and \ \hat{k}, \hat{i} . \hat{i}=\hat{j} \cdot \hat{j}=\hat{k} \cdot \hat{k}=1\)
7. In the case of orthogonal unit vectors, \(\hat{i}, \hat{j} \ and \ \hat{k}, \hat{i}, \hat{j}=\hat{j}, \hat{k}=\hat{k} \cdot \hat{k}=0\)
8. Work is the example for dot product.

Th e time taken by the sound to come back as echo is 2t = 4 \(\Rightarrow\) t = 2 s
\(\therefore\) Th e distance is d = vt =(343 m s^-1)(2 s) = 686 m.

System gains heat Q is positive
System loses heat Q is negative
Work done on the system W is negative
Work done by the system W is positive

\(\overrightarrow{r}=7\hat{i}+4\hat{j}-2\hat{k}\)
\(\overrightarrow{F}=4\hat{i}-3\hat{j}+5\hat{k}\)
Torque,\(\overrightarrow{\tau}=\overrightarrow{r}\times \overrightarrow{F}\)
\(\overrightarrow{\tau}=\left| \begin{matrix} \hat { i } & \hat {j } & \hat {k } \\ 7 &4 & -2 \\ 4 & -3 & 5 \end{matrix} \right| \)
\(\overrightarrow{\tau}=\hat{i}(20-6)-\hat{j}(35+8)+\hat{k}(-21-16)\)
\(\overrightarrow {\tau}=(14\hat{i}-43\hat{j}-37\hat{k})Nm\)

In Physics, work is said to be done by the force when applied on a body displaces it. To do work, energy is required. But, generally work refers to both physical and mental work. In fact, any activity can be called as work.

If an object is thrown vertically upward, the direction of motion is upward, but gravitational force is downward.

Least count of screw gauge
\(={{Pitch}\over{No. of\ divisions}}\)
\(={{1}\over{100}}=0.01\ mm\) (or) 0.001 cm
Least count of vernier calipers
\(=1MSD-1-1VSD=(1-19/20)MSD\)
\(={{1}\over{20}}\) = 0.05 cm.
So screw gauge is more precise than vernier.