Question

An amusement park ride called "The spinning Terror" is a large vertical drum which spins so fast that everyone stays pinned against the wall when the floor drops away from, the minimum
A) Angular velocity is \[{\omega _{min}} = \sqrt {\dfrac{g}{{\mu R}}} \] for everyone to stay inside.
B) Linear velocity is \[{v_{min}} = \sqrt {\dfrac{g}{{\mu R}}} \] for everyone to stay Inside.
C) Angular velocity is \[{\omega _{min}} = \sqrt {\dfrac{{gR}}{\mu }} \] for everyone to stay Inside.
D) Linear velocity is \[{v_{min}} = \sqrt {\dfrac{{gR}}{\mu }} \] for everyone to stay Inside.

Answer 1

Hint: Linear velocity is defined as the rate of change of the displacement to the time, whenever the body is moving along a straight path. Angular velocity is defined as the rate of change of the angular position of a rotating body.

Complete step by step solution:
Given that it is a vertical drum.

We know that $mg = \mu N\_\_\_\_\_\_\_\_\_\_\left( 1 \right)$
From the diagram, $N = m{\omega ^2}_{min}R$
Substituting the value of N in equation 1, we get
$ \Rightarrow mg = \mu m{\omega ^2}_{min}R$
\[\therefore {\omega _{min}} = \sqrt {\dfrac{g}{{\mu R}}} \]
Also, we know that ${v_{min}} = R{\omega _{min}}$
Hence substituting the value of \[{\omega _{min}}\], we get
$ \Rightarrow {v_{min}} = R\sqrt {\dfrac{g}{{\mu R}}} = \sqrt {\dfrac{{gR}}{\mu }} $

Hence the correct options are A and D.

Note: 1. Angular velocity is the change in angle over time whereas linear velocity is the speed in a straight line. The unit of linear velocity is m/s and the unit of angular velocity is rad/s.
2. When the object is coming in the same direction, in which the gravity is acting then the acceleration due to gravity is positive.
3. The rate of change of velocity is known as acceleration and the rate of change angular velocity is known as angular acceleration. Angular velocity and angular speed are both the same.
4. The acceleration gained by the body due to gravity is called acceleration due to gravity. The acceleration due to gravity depends on the mass of the earth and the radius of the earth. It is denoted by the symbol g. The acceleration due to gravity is maximum at the poles and is minimum at the equator. When a body is going up, the acceleration is negative as the speed is decreasing.