The radius of gyration of a body about an axis is the distance from the axis where:
the body's mass is concentrated
the body's mass produces the maximum torque
the body's moment of inertia is maximum
the body's moment of inertia is minimum
The question asks for the definition of the radius of gyration. We need to recall the concept of moment of inertia and how the radius of gyration is defined in relation to it.
Correct Option: A) the body's mass is concentrated. This aligns directly with the definition of the radius of gyration, where the entire mass of the body is imagined to be concentrated at this distance to produce the same moment of inertia.
For a particle of a rigid body rotating about a fixed axis, at a fixed distance r from the axis, the angular velocity is directly proportional to:
the radius of the body
the moment of inertia of the body
the linear velocity of the particle
the mass of the body
This question asks about the relationship between angular velocity and other physical quantities for a particle in a rigid body rotating about a fixed axis. We need to recall the definitions and relationships between linear velocity and angular velocity in rotational motion.
Correct Option: C) the linear velocity of the particle
As derived, for a particle at a fixed distance \(r\) from the axis of rotation, the angular velocity \(\omega\) is directly proportional to its linear velocity \(v\).
The moment of inertia, often denoted by \(I\), is a measure of an object's resistance to changes in its rotational motion. It is analogous to mass in linear motion. To determine its unit, we can recall its definition or formula.
C) kg m² — This is the correct unit for moment of inertia, derived directly from its definition as mass times the square of the distance from the axis of rotation.
The question asks for the correct expression for torque acting on a particle. Torque is a rotational equivalent of force and is a measure of the force that can cause an object to rotate about an axis. We need to recall its definition and formula.
B) Force \(\times\) perpendicular distance from the axis of rotation. This correctly represents the magnitude of torque, where the perpendicular distance is the lever arm.
To determine how the kinetic energy changes when the angular velocity is doubled, we need to recall the formula for rotational kinetic energy and analyze its dependence on angular velocity.
where \(E_k\) is the rotational kinetic energy, \(I\) is the moment of inertia, and \(\omega\) is the angular velocity.
Now, the angular velocity is doubled, so the new angular velocity \(\omega_2 = 2\omega_1\). Let the new kinetic energy be \(E_{k2}\). Substituting the new angular velocity into the formula:
\[ E_{k2} = \frac{1}{2} I (2\omega_1)^2 \]\[ E_{k2} = \frac{1}{2} I (4\omega_1^2) \]\[ E_{k2} = 4 \left( \frac{1}{2} I \omega_1^2 \right) \]This means the new kinetic energy is four times the initial kinetic energy.
A) Four times — When the angular velocity is doubled, the kinetic energy becomes four times its original value because kinetic energy is proportional to the square of the angular velocity (\(E_k \propto \omega^2\)).
To determine the direction of acceleration in uniform circular motion, we need to understand the nature of velocity and acceleration in this type of motion. Uniform circular motion implies constant speed but continuously changing velocity direction, which necessitates an acceleration.
B) Radial, towards the center of the circle — In uniform circular motion, the acceleration (centripetal acceleration) is always directed radially inwards, towards the center of the circle. This continuous change in the direction of velocity is what keeps the object moving in a circle.