Question

Question: -An ice skater is executing a figure eight, consisting of two identically shaped, tangent circular paths. Throughout the first loop she increases her speed uniformly, and during the second loop she moves at a constant speed. Draw a motion diagram showing her velocity and acceleration vectors at several points along the path of motion.

Short answer

The diagrams are drawn in the below steps

Step-by-step solution

Concept

The motion of a body on a circular path with a constant speed is known as Uniform Circular Motion. In this type of motion, the only acceleration that acts on the body is the centripetal acceleration, that is always directed towards the center of the circular path.

Velocity and acceleration vector in the first loop

During the first loop, the tangential acceleration will increase the skater's speed as follows:
$v=(2a_{t}d{)}^{\frac{1}{2}}$

Here, the distance is

The radial component of the acceleration’s magnitude is:
$a_r=\frac{v^2}{r}$

In the above equation the velocity is and the radius of circular path is r.

The skater's velocity rises throughout the first loop, and as a result of the aforementioned calculation, the radial acceleration rises as well, pointing toward the circular path's centre.

The angle formed by the tangential line's resultant acceleration is,
$\tan \theta =\frac{a_r}{a_t}$

Here, and is increasing and is constant.
The tangential acceleration will remain constant while the radial acceleration will increase. The following are the directions of the components of acceleration and the resultant acceleration in the motion:

is the resultant acceleration in the above figure.

Velocity and acceleration vector in the second loop

The skater maintains a consistent speed throughout the second loop. As a result, the

tangential acceleration for the second loop is zero.

$a_t=0{\text{m/s}}^{\text{2}}$

Here also is constant in second loop.

So, the direction of the acceleration of the skater is shown in the figure below: