MHT-CET 2020 (13-Oct Shift 1)

In U.C.M., when time interval $\delta \to 0$ , the angle between change in velocity $\left(\delta v\right)$ and linear velocity ($v$) will be

A particle performing $\mathrm{U}.\mathrm{C}.\mathrm{M}.$ of radius $\frac{\pi }{2} \mathrm{m}$ makes $x$ revolutions in time $t$. Its tangential velocity is

A mass $m$ is tied to one end of a spring and whirled in a horizontal circle with constant angular velocity. The elongation in the spring is $1 \mathrm{cm}$. If the angular speed is doubled, the elongation in the spring is $6 \mathrm{cm}$. The original length of the spring is

Two cars of masses $m_1$, and $m_2$ are moving in the circles of radii $r_1$ and $r_2$ respectively. Their angular speeds ${{}^{\text{'}}\omega _1}^{\text{'}}$ and ${{}^{\text{'}}\omega _2}^{\text{'}}$ are such that they both complete one revolution in the same time $t$. The ratio of linear speed of $m_1$' to the linear speed of $m_2$ is