Category: HC Verma Part 1: Mechanics

Problem Statement Show that $\vec{A}\cdot\vec{A} = A^2$ (the square of the magnitude). Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units…

Problem Statement Solve the kinematics problem: A particle $A$ moves with velocity $\vec{v}_A = 4\hat{i}$ m/s and particle $B$ moves with $\vec{v}_B = 3\hat{j}$ m/s. Find the velocity of $A$ relative to $B$. $\vec{v}_{AB} = \vec{v}_A – \vec{v}_B$ Step 1: $\vec{v}_{AB} = 4\hat{i} – 3\hat{j}$ m/s. Step 2: $|\vec{v}_{AB}| = \sqrt{16+9} = 5$ m/s. $$\…

Problem Statement If $\vec{A} = 5\hat{i} + 3\hat{j}$ and $\vec{B} = 2\hat{i} – 4\hat{j}$, find $\vec{A} – \vec{B}$. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving…

Problem Statement Solve the work-energy problem: A force $F = 2x$ N acts on a particle. Find the work done as it moves from $x = 1$ m to $x = 3$ m. $W = \int_{x_1}^{x_2} F\,dx$ Step 1: $W = \displaystyle\int_1^3 2x\,dx = \left[x^2\right]_1^3 = 9 – 1 = 8$ J. $$\boxed{W = 8\text{…

Problem Statement Solve the kinematics problem: A particle has acceleration $a = 2t$ m/s². If $v = 0$ at $t = 0$, find $v$ at $t = 3$ s. $v = v_0 + \int_0^t a\,dt$ Step 1: $v = \displaystyle\int_0^t 2t’\,dt’ = t^2$. Step 2: At $t = 3$: $v = 9$ m/s. $$\boxed{v =…

Problem Statement Find the slope of the curve $y = x^3$ at $x = 2$. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful…

Problem Statement Find $\dfrac{d}{dt}\left[\sin^2(\omega t)\right]$. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions. See the step-by-step solution…

Problem Statement If $x = A\cos(\omega t + \phi)$, show that $\ddot{x} = -\omega^2 x$. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful…

Problem Statement Evaluate $\displaystyle\int_0^T \sin(\omega t)\,dt$. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions. See the step-by-step…

Problem Statement If a vector $\vec{F}$ makes angles $\alpha$, $\beta$, $\gamma$ with the $x$-, $y$-, $z$-axes respectively, show that $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant…