Laws of Motion

The time taken to slide down the rough surface is twice that of the smooth surface: $t_{\text{rough}} = 2t_{\text{smooth}}$.

The acceleration on the smooth surface is given by:

$ a_{\text{smooth}} = g \sin \theta $

The time to slide down an inclined plane is inversely proportional to the square root of the acceleration:

$ t \propto \frac{1}{\sqrt{a}} \Rightarrow t_{\text{smooth}} \propto \frac{1}{\sqrt{g \sin \theta}} $

The acceleration on the rough surface is:

$ a_{\text{rough}} = g \sin \theta - \mu_k g \cos \theta $

Relating the times on both surfaces, we have:

$ \frac{t_{\text{rough}}}{t_{\text{smooth}}} = \frac{\sqrt{\sin \theta}}{\sqrt{\sin \theta - \mu_k \cos \theta}} = 2 $

Squaring both sides results in:

$ \frac{\sin \theta}{\sin \theta - \mu_k \cos \theta} = 4 $

Substituting $\theta = 45^{\circ}$ where $\sin \theta = \cos \theta = \frac{1}{\sqrt{2}}$, we find:

$ \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}} - \mu_k \times \frac{1}{\sqrt{2}}} = 4 $

Simplifying further:

$ 1 - \mu_k = \frac{1}{4} $

Solving for $\mu_k$:

$ \mu_k = \frac{3}{4} = 0.75 $

Thus, the coefficient of kinetic friction is $0.75$.

$T-M g \sin 30=M a$

$\begin{aligned} & 30-5 \times 10 \times \frac{1}{2}=5 a \\ & a=1 \mathrm{~m} / \mathrm{s}^2 \end{aligned}$

$ \begin{aligned} & F=\left(M_1+M_2\right) a \\ & a=\frac{10}{2+3}=2 \mathrm{~ms}^{-2} \\ & F^{\prime}=M_2(2)=3 \times 2 \mathrm{~N}=6 \mathrm{~N} \end{aligned} $

When a particle moves along a circular path with a uniform speed, it is important to understand the motion characteristics in terms of both velocity and acceleration. Velocity is a vector quantity which means it has both magnitude and direction, while acceleration is the rate of change of velocity with respect to time.

First, let's analyze the velocity:

Although the speed (magnitude of the velocity vector) remains constant in uniform circular motion, the direction of the velocity vector continuously changes as the particle progresses along the circle. Since velocity includes both the magnitude and the direction, any change in either results in a change in velocity. Consequently, in uniform circular motion, the velocity of the particle is not constant but varies due to the continuous change in direction.

Next, consider the acceleration:

In circular motion, there is always an acceleration directed towards the center of the circle, known as centripetal acceleration. This acceleration is responsible for changing the direction of the velocity vector, thereby keeping the particle moving in a circle, despite the speed being constant. The formula for centripetal acceleration is:

$ a_c = \frac{v^2}{r} $

where $ v $ is the speed of the particle and $ r $ is the radius of the circle. This acceleration is always directed towards the center of the circle and varies with the square of the speed and inversely with the radius of the circle.

Therefore, given that both the velocity and acceleration change, the correct choice is:

Option D - Varying velocity and varying acceleration

It states that in uniform circular motion, both the velocity and acceleration of the particle vary – velocity due to continuous changes in direction and acceleration due to its consistent inward (centripetal) direction towards the center of the circle.

$\mu=\frac{1}{\sqrt3}$

$\tan30^\circ=\frac{1}{\sqrt3}$

as $\mu=\tan\theta$

the block is at rest and net force on it must be zero.

$\vec{V}_{\mathrm{i}}=(\mathrm{V})$ Southward

$\overrightarrow{\mathrm{V}}_{\mathrm{F}}=(\mathrm{V})$ Eastward

$\overrightarrow{\Delta \mathrm{V}}=\overrightarrow{\mathrm{V}}_{\mathrm{F}}-\overrightarrow{\mathrm{V}}_{\mathrm{i}}$

= Along North - East

To find the maximum acceleration ($a_{\max}$) of the car that allows a body to stay stationary relative to the car, we use the concept of static friction. Static friction ($F_s$) is what keeps the body from sliding on the car's floor. It acts in the opposite direction of the potential movement of the body.

The maximum static friction force is given by $F_{s_{\max}} = \mu_s \times N$ where $\mu_s$ is the coefficient of static friction and $N$ is the normal force. In this scenario, the normal force is equal to the gravitational force on the body ($mg$), where $m$ is the mass of the body and $g$ is the acceleration due to gravity.

Since the maximum force of static friction equals the product of the mass and the maximum acceleration ($ma_{\max}$), we have:

$\mu_s mg = ma_{\max}$

By canceling out the mass $m$ on both sides, we get:

$a_{\max} = \mu_s g$

Substituting the given values ($\mu_s = 0.15$ and $g = 10 \,\mathrm{m/s}^2$):

$a_{\max} = 0.15 \times 10 = 1.5 \,\mathrm{m/s}^2$

Therefore, the maximum acceleration of the car to ensure the body remains stationary with respect to the car's floor is $1.5 \,\mathrm{m/s}^2$.

$B y v^{2}=u^{2}+2 a s$

$\left(\frac{u}{3}\right)^{2}=u^{2}-2 a x$

$2 a x=u^{2}-\frac{u^{2}}{9}$

$2 \mathrm{ax}=\frac{8 \mathrm{u}^{2}}{9}$ ..... (1)

Similarly from starting

$v^{2}=u^{2}+2 a x$

$0=u^{2}-2 a x_{2}$

$2 \mathrm{ax}_{2}=\mathrm{u}^{2}$ ..... (2)

$\mathrm{By}~(1) /(2)$

$\frac{x}{x_{2}}=\frac{8}{9}$

$\frac{24}{x_{2}}=\frac{8}{9}$

$\mathrm{x}_{2}=27 \mathrm{~cm}$

Given, $\overrightarrow F = 2\widehat i + \widehat j - \widehat k$ and $\overrightarrow r = 3\widehat i + 2\widehat j - 2\widehat k$

$\overrightarrow F \,.\,\overrightarrow r = 2(3) + 2(1) + ( - 2)( - 1)$

$ = 6 + 2 + 2 = 10$

$\overrightarrow F \times \overrightarrow r = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 2 & 1 & { - 1} \cr 3 & 2 & { - 2} \cr } } \right|$

$ = \widehat i( - 2 + 2) - \widehat j( - 4 + 3) + \widehat k(4 - 3)$

$ \Rightarrow \overrightarrow F \times \overrightarrow r = \widehat j + \widehat k$

$|\overrightarrow F \times \overrightarrow r | = \sqrt {{1^2} + {1^2}} = \sqrt 2 $

Assuming all three blocks as system

Hence, acceleration of system

$a = {{60 - 30 - 18} \over 6} = {{12} \over 6} = 2$ m/s²

$a = 2$ m/s², upward along the incline.

Now, for 1 kg block

$ \Rightarrow {N_{12}} - 18 - 5 = 1 \times 2$

$ \Rightarrow {N_{12}} - 23 = 2$

$ \Rightarrow {N_{12}} = 25$ N