The angular displacement of the rod is defined as θ=(3)/(20) t^2 where θis in radian and t is in

step 1 Hello students. In this question we have angular displacement of the rod theta equals to 3 by 20

The angular displacement of the rod is defined as θ=(3)/(20)...

The angular displacement of the rod is defined as $\theta=\frac{3}{20} t^{2}$ where $\theta$ is in radian and $t$ is in second. The collar $B$ slides along the rod in such a way that its distance from $O$ is, $r=0.9-0.12 t^{2}$ where $r$ is in metre and $t$ is in second. The velocity of collar at $\theta=30^{\circ}$ is: (a) $0.45 \mathrm{~m} / \mathrm{s}$ (b) $0.48 \mathrm{~m} / \mathrm{s}$ (c) $0.52 \mathrm{~m} / \mathrm{s}$ (d) $0.27 \mathrm{~m} / \mathrm{s}$

The angular displacement of the rod is defined as $\theta=\frac{3}{20} t^{2}$ where $\theta$ is in radian and $t$ is in second. The collar $B$ slides along the rod in such a way that its distance from $O$ is, $r=0.9-0.12 t^{2}$ where $r$ is in metre and $t$ is in second. The velocity of collar at $\theta=30^{\circ}$ is:
(a) $0.45 \mathrm{~m} / \mathrm{s}$
(b) $0.48 \mathrm{~m} / \mathrm{s}$
(c) $0.52 \mathrm{~m} / \mathrm{s}$
(d) $0.27 \mathrm{~m} / \mathrm{s}$

Key Concepts

Rotational Kinematics

This concept involves describing the motion of objects in rotation through angular displacement, velocity, and acceleration. In problems involving rotating bodies, angular quantities are given as functions of time, and understanding these relationships is key to analyzing the motion and correlating angular measurements to the dynamic behavior of the system.

Relative Motion in Polar Coordinates

When an object moves along a path that is rotating (or defined in a circular coordinate system), its velocity can be broken down into radial and tangential components. The radial component is due to the change in distance from a fixed point, while the tangential component arises from the rotational motion of the system. The overall velocity is obtained by combining these two perpendicular components.

Differentiation of Time-Dependent Functions

In dynamics problems where position functions are given as functions of time, differentiation is used to calculate the corresponding velocity or acceleration components. For example, differentiating the expressions for radial distance and angular displacement with respect to time yields the radial and tangential velocities, which are essential for determining the magnitude of the total velocity.

Unit Conversion

Often in physics problems, angles may be specified in different units such as degrees or radians. Converting between these units is vital for consistency, especially when formulas require a specific unit (typically radians) for correct application. This ensures that the computed time or other variables derived from angular relationships are accurate.

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