Question

Three people are standing on a horizontally rotating platform in an amusement park. One person is almost at the edge, the second one is (3/5)R from the center, and the third is (1/2)R from the center.



1-If the period of rotation of the first person is T1 = T, find in terms of Tthe period of rotation of the second person.


2- If the period of rotation of the first person is T1 = T, find in terms of T the period of rotation of the third person.


3-If the speed of the first person is v1 = v find in terms of v the speed of the second person


4- If the speed of the first person is v1 = v find in terms of v the speed of the third person.


5- If the radial acceleration of the first person is a1 = a, find in terms of athe radial acceleration of the second person.


6- If the radial acceleration of the first person is a1 = a, find in terms of athe radial acceleration of the third person.

Answer 1

Answer:

1- Period is the time for which one full rotation is completed. Regardless of their positions on the platform, periods of all three are the same. T2 = T.

2- Similarly, T3 = T.

3- The platform is making rotational motion. So, the relation between the angular velocity and the linear velocity is

$v = \omega R$

For all the people, angular velocity is the same. Their linear velocities are different.

$\omega = \frac{v_1}{R} = \frac{v_2}{3R/5}\\v_2 = \frac{3v}{5}$

4- Similarly,

$\omega = \frac{v_1}{R} = \frac{v_2}{R/2}\\v_2 = \frac{v}{2}$

5- Radial acceleration in constant circular motion is

$a_{rad} = \frac{v^2}{R}$

For the second person:

$a_2 = \frac{v_2^2}{3R/5} = \frac{(3v/5)^2}{3R/5} = \frac{9v^2/25}{3R/5} = \frac{3v^2}{5R} = 3a/5$

6- Similarly,

$a_3 = \frac{v_3^2}{R/2} = \frac{(v/2)^2}{R/2} = \frac{v^2/4}{R/2} = \frac{v^2}{2R} = a/2$

Explanation:

As a result, the period is same for every object on the rotating platform, as they all complete their revolutions at the same time. Their speed and radial acceleration is different according to their distance to the center.

Answer 2

The rotational kinematics relations allow to find the results for the questions about the movement of the three people on the turntable are:

      1 and 2) All periods are equal, T₂ = T and T₃ = T.

       3) The linear velocity of the 2nd person is: v₂ = $\frac{3}{5} \ v$  

       4) The linear velocity of the 3rd person is: v₃ = ½ v

       5) The linear acceleration of the 2nd person is: a₂ = $\frac{3}{5} \ a$  

      6) The linear acceleration of the 3rd person: a₃ = ½ a

Rotational kinematics studies the rotational motion of bodies looking for relationships between angular position, angular velocity, and angular acceleration.

In the case where the angular accleration is zero, the expression for the velocity is:

        $w = \frac{\Delta \theta }{\Delta t}$  

Where w is the angular velocity and Δw and Δt are the variation in angle t over time.

1 and 2)

Indicates that people are on a turntable, the period is when we have a complete rotation θ = 2π rad in time, therefore the period and the angular velocity are related.

          $w= \frac{2\pi }{T} \\T = \frac{2\pi }{w}$  

In the apparatus of parks the angular velocity is constant and we see that it does not depend on the radius, therefore the period for all the people is the same.

         T = T₁ = T₂

3) They indicate that the speed of the 1 person who is in the position r=R on the plate is v, let's calculate the speed for the 2 person who is in the position  r = $\frac{3}{5} \ R$  

Linear and angular variables are related.

        v = w r

Let's substitute for the 1st person.

        v = w R

For the 2nd person.

        v₂= w ( $\frac{3}{5} R$)

We solve these two equations.

         $v_2 = \frac{3}{5} \ v$  

4) We carry out the same calculation for the 3rd person.

         v₃ = w ½ R

We solve the two equations.

          v₃ = ½ v

5) Ask for radial acceleration.  

The relationship between radial and angular acceleration is.

           a = α R

We substitute for the 1st person.

          a = α R

For the second person.

          a₂ = α ( $\frac{3}{5} R$)

We solve the two equations

         a₂ = $\frac{3}{5} \ a$

6) Ask the radial acceleration of the 3rd person.

We substitute.

         a₃ = α (½ R)

We solve.

        a₃ = ½ a

In conclusion, using the rotational kinematics relations we can find the results for the questions about the movement of the three people on the turntable are

      1 and 2) All periods are equal, T₂ = T and T₃ = T.

       3) The linear velocity of the 2nd person is: v₂ = $\frac{3}{5} \ v$  

       4) The linear velocity of the 3rd person is: v₃ = ½ v

       5) The linear acceleration of the 2nd person is: a₂ = $\frac{3}{5} \ a$  

      6) The linear acceleration of the 3rd person: a₃ = ½ a

Learn more about rotational kinematics here:  brainly.com/question/14524058