Dynamics of rotational motions – problems and solutions

1. A pulley with the moment of inertia I = 2/5 MR^{2} has a mass of 2-kg. If the moment of force on the pulley is 4 N.m then what is the linear acceleration of the pulley. Acceleration due to gravity is g = 10 m.s^{-2}.

__Known ____:__

The moment of inertia of the pulley (I) = 2/5 MR^{2}

Mass of pulley (M) = 2 kg

Moment of force (τ) = 4 Nm

Acceleration due to gravity (g) = 10 m.s^{-2}

Radius of pulley (R) = 20 cm = 0.2 m

__Wanted :__ The linear acceleration (a)

__Solution :__

The moment of inertia of pulley (I) :

I = 2/5 MR^{2 }= 2/5 (2)(0.2)^{2} = 2/5 (2)(0.04) = 2/5 (0.08) = 0.16 / 5 = 0.032 kg m^{2}

The angular acceleration (α) :

τ = I α

α = τ/I = 4 / 0.032 = 125 rad.s^{-2}

The linear acceleration (a) :

a = R α = (0.2)(125) = 25 m.s^{-2}

The linear acceleration of the edge of pulley is 25 m.s^{-2}.

2. Acceleration of pulley is 2 ms^{-2}. Acceleration due to gravity is g = 10 m.s^{-2}. What is the moment of inertia of the pulley.

__Known :__

The linear acceleration of the edge of pulley (a) = 2 ms^{-2}

Acceleration due to gravity (g) = 10 m.s^{-2}

Radius of pulley (R) = 10 cm = 0.1 m

Object’s weight (w) = m g = (4)(10) = 40 N

__Wanted :__ The moment of inertia of pulley

__Solution :__

The moment of force (τ) :

τ = F R = w R = (40)(0.1) = 4 Nm

The angular acceleration of the pulley (α) :

a = R α

α = a/R = 2 / 0.1 = 20 rad.s^{-2 }

The moment of inertia of the pulley (I) :

τ = I α

I = τ/α = 4/20 = 0.2 kg m^{2}

3. The pulley accelerated at 1 ms^{-2}. If radius of pulley is 10 cm,what is the moment of inertia of the pulley.

__Known :__

The linear acceleration of the edge of pulley (a) = 1 ms^{-2}

Acceleration due to gravity (g) = 10 m.s^{-2}

Radius of pulley (R) = 10 cm = 0.1 m

Force (F) = 4 N

__Wanted :__ The moment of inertia of the pulley

__Solution :__

The moment of force (τ) :

τ = F R = (4)(0.1) = 0.4 Nm

The angular acceleration of pulley (α) :

a = R α

α = a/R = 1 / 0.1 = 10 rad.s^{-2 }

The moment of inertia of pulley (I) :

τ = I α

I = τ/α = 0.4 / 10 = 0.04 kg m^{2}

**What is the rotational equivalent of mass in linear motion?****Answer:**The rotational equivalent of mass in linear motion is the moment of inertia, often denoted as $I$.

**How does torque relate to rotational motion, and how does it compare to force in linear motion?****Answer:**Torque is the rotational equivalent of force. While force causes linear acceleration in objects, torque causes angular acceleration in rotating bodies. Torque is calculated as the product of the force applied and the perpendicular distance from the axis of rotation to the point of application of the force.

**What is the relationship between angular acceleration, torque, and moment of inertia?****Answer:**The relationship is given by Newton’s second law for rotation: $τ=Iα$, where $τ$ is the torque, $I$ is the moment of inertia, and $α$ is the angular acceleration.

**If the moment of inertia of a rotating object increases while the external torque remains constant, what happens to the angular acceleration?****Answer:**If the moment of inertia increases and the external torque remains constant, the angular acceleration will decrease.

**How are the concepts of linear momentum and angular momentum analogous in linear and rotational motion?****Answer:**Linear momentum (given by $p=mv$, where $m$ is mass and $v$ is velocity) is the product of mass and linear velocity, whereas angular momentum (given by $L=Iω$, where $I$ is the moment of inertia and $ω$ is the angular velocity) is the product of the moment of inertia and angular velocity.

**What is meant by the conservation of angular momentum?****Answer:**The conservation of angular momentum states that in the absence of external torques, the total angular momentum of a closed system remains constant.

**Why do figure skaters spin faster when they tuck their arms in during a spin?****Answer:**When figure skaters tuck their arms in, they decrease their moment of inertia. Due to the conservation of angular momentum, when the moment of inertia decreases, the angular velocity (or rate of spin) must increase to keep the angular momentum constant.

**How does the rotational kinetic energy of an object compare to its linear kinetic energy?****Answer:**The rotational kinetic energy (given by 1/2 $ Iω$) is analogous to linear kinetic energy (given by 1/2 $ mv$). The former is associated with an object’s rotation, and the latter with its linear motion.

**Can an object have both rotational and linear kinetic energy simultaneously?****Answer:**Yes, an object can have both rotational and linear kinetic energy. For example, a rolling wheel has rotational kinetic energy due to its spin and linear kinetic energy due to its forward motion.

**If the radius of a rotating wheel doubles while keeping its angular velocity constant, how does its moment of inertia change?**

**Answer:**The moment of inertia of a solid disk is given by 1/2 $ mr$, where $m$ is the mass and $r$ is the radius. If the radius doubles and the mass remains constant, the moment of inertia will increase by a factor of four.