91Ó°ÊÓ

Radial acceleration, measured in radians per square second, is the acceleration that is moving in the direction of the center and is caused by the centripetal force. Tangential acceleration occurs when a body or object travels at a non-uniform speed.

The forces and acceleration are shown in the figure below.

In the figure, the given variables are as follows.

The length is L is the length of the rod, T is the tension on the rod, $\mathrm{θ}$is an angle, a is the acceleration, and w is the weight of the sphere.

The acceleration is define from the figure as below.

localid="1668146044093" $a=\sqrt{{a_{tan}}^2+2_{r}a{d}^2}$ ….. (1)

Here,

The radial acceleration, localid="1668146047525" $a_{rad}=L{\mathrm{Ó\neg }}^2$

The angular frequency islocalid="1668146055124" $\mathrm{Ó\neg }$.

The tangential acceleration, localid="1668146051490" $a_{tan}=L\mathrm{Î\pm }$

The angular acceleration is $\mathrm{Î\pm }$.

In this case, the radial acceleration is zero. Therefore,

localid="1668146070600" $a_{rad}=0$

Substitute this into equation (1).

localid="1668146076940" $$\begin{gathered} a=a_{\tan } \\ =g\sin \mathrm{θ} \end{gathered}$$

Hence, the acceleration of the sphere is localid="1668146084785" $g\sin \mathrm{θ}$.

The forces and acceleration are shown in Figure below.

The forces and acceleration are shown in Figure below.

Energy conservation method is used to calculate the velocity. So,

$$\begin{gathered} U=K \\ mgL(1-\cos \mathrm{θ})=\frac{1}{2}m{v}^2 \end{gathered}$$

Here, is the potential energy, is the kinetic energy. is the mass, is the acceleration due to gravity, is the length, is the angle, and is the velocity.

Solve the above equation for the velocity and obtain the expression as,

$v=\sqrt{2gL(1-\cos \mathrm{θ})}$

Hence, the required velocity expression is $v=\sqrt{2gL(1-\cos \mathrm{θ})}$.