Test: Rotational Work and Energy - EmSAT Achieve MCQ

CONCEPT:

• Potential Energy: Potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
• Potential energy formula: The formula for potential energy depends on the force acting on the two objects. For the gravitational force the formula is:

where m is the mass in kilograms, g is the acceleration due to gravity, h is the height in meters.

• Rotational Kinetic Energy: The kinetic energy of a rotating body can be compared to the linear kinetic energy and described in terms of angular velocity. 
  • Rotational energy occurs due to the object's rotation and is a part of its total kinetic energy. If the rotational energy is considered separately across an object's axis of rotation, the moment of inertia observed.
    • Rotational energy is also known as angular kinetic energy defined as: "The kinetic energy due to the rotation of an object and is part of its total kinetic energy".
  • Rotational kinetic energy is directly proportional to the rotational inertia and the square of the magnitude of the angular velocity. A rolling object has both translational and rotational kinetic energy.

K_R is the Rotational Kinetic energy, I is the Moment of Inertia, ω is the angular velocity

​CALCULATION:

Using conservation of Energy:

Potential energy = Rotational kinetic energy

Hence option 2 is the correct answer.

CONCEPT:

Kinetic energy (K.E): 

• The energy possessed by a body by virtue of its motion is called kinetic energy.
• The expression for kinetic energy is
  

Where m = mass of the body and v = velocity of the body

Rotational Kinetic energy (KE):

• The energy, which a body has by virtue of its rotational motion, is called rotational kinetic energy.
• A body rotating about a fixed axis possesses kinetic energy because its constituent particles are in motion, even though the body as a whole remains in place.
• Mathematically rotational kinetic energy can be written as
  
  Where I = moment of inertia and ω = angular velocity.
  Condition of rolling is
  ⇒ V = rω 
  Where ω = angular velocity and V = linear velocity 

CALCULATION: 

Given - mass (m) = 10 kg and velocity (V) = 2 m/s

• Moment of inertia of thin circular ring is 

⇒ I = mR²

• Work (in joules) needs to be done to stop it is
  

CONCEPT:

• Rotational motion: When a block is moving about a fixed axis on a circular path then this type of motion is called rotational motion.
• Torque (τ): It is the twisting force that tends to cause rotation.
  • The point where the object rotates is known as the axis of rotation. 

Mathematically it is written as,

τ = r F sin θ 

Where r is the distance between the axis of rotation and point of application of force, F is force and θ is the angle between r and F.

• Rotational power (P): The rate of work done by torque is called power.

The power associated with torque is given by the product of torque and angular velocity of the body about an axis of rotation i.e.,

⇒ P = τ ω 

Where τ = torque and ω = angular velocity

Angular momentum (L) = Rotational inertia (I) × Angular velocity (ω)

EXPLANATION:

Work done by the torque (W) = Torque (τ) × angular displacement (θ)

• The product of torque and angular displacement is work. So option 2 is correct.

EXTRA POINTS:

In rotational dynamics

• Moment of inertia is the analogue of mass
• Angular velocity is analogue of linear velocity
• Angular acceleration is analogue of linear acceleration

Thus, in linear motion mass x velocity = momentum

The analogues of the moment of inertia x angular velocity = angular momentum

CONCEPT:

• When a body moves in a linear direction, it possesses kinetic energy.​
• The Kinetic Energy of an object due to its linear speed is given by 

​​E = 1/2 (m × v²) 

where m is the mass of a body and v is the speed.

• When a body does the rotational motion, it possesses rotational kinetic energy.

​​E = 1/2 (I × ω²) 

where I is the moment of inertia of the body, and ω is the angular velocity.

• When a body is at a certain height, it has potential energy.

PE = m g h

where m is the mass of a body, g is gravitational acceleration, and h is the height from the earth's surface.

EXPLANATION:

• Kinetic energy: Since the body is moving in a linear direction, so it will have kinetic energy due to its velocity.
• Rotational energy: Along with linear velocity, the body is also revolving i.e. rotating, so it will have rotation energy.
• Potential energy: Since the body is at a certain height, it will have potential energy.
• Hence the body will have all three Kinetic energy, Rotational energy, and potential energy.
• So the correct answer is option 4.

CONCEPT:

• Work-energy theorem: It states that work done by a force acting on a body is equal to the change in the kinetic energy of the body i.e.,

W = K_f - K_i

Where v = final velocity, u = initial velocity and m = mass of the body

CALCULATION:

Given - mass (m) = 1500 kg, initial velocity (u) = 36 km/hr = 10 m/s and final velocity (v) = 72 km/hr = 20 m/s

• The work done to increase the velocity of a 1500 kg car from 36 km/h to 72 km/h is

CONCEPT:

Kinetic energy (KE):

• The energy, which a body has by virtue of its rotational motion, is called rotational kinetic energy.
• A body rotating about a fixed axis possesses kinetic energy because its constituent particles are in motion, even though the body as a whole remains in place.
• Mathematically rotational kinetic energy can be written as –

Where I = moment of inertia and ω = angular velocity.

• The moment of inertia of a solid cylinder of mass 'M' and radius 'R' about the axis of the cylinder

CALCULATION:

Given -

M = 10 kg, R = 0.5 m and ω = 20 rad/s

Mathematically rotational kinetic energy can be written as 

Important Point

The correct answer is Kinetic Energy.

• The kinetic energy of an object is defined as the energy that it possesses due to its motion.

Key Points

• Thus, an object that has motion possesses kinetic energy.
• The amount of kinetic energy that an object has depends upon two variables i.e. the mass (m) of the object and the speed (v) of the object.
• A bullet moves with a large velocity and as such possesses a lot of kinetic energy.
• The work done in piercing the target is derived from the kinetic energy of the bullet.

CONCEPT:

• Rotational kinetic energy: The energy, which a body has by virtue of its rotational motion, is called rotational kinetic energy.
• A body rotating about a fixed axis possesses kinetic energy because its constituent particles are in motion, even though the body as a whole remains in place.
• Mathematically rotational kinetic energy can be written as -

Where I = moment of inertia and ω = angular velocity.

EXPLANATION:

• From above it is clear that,  represents rotational kinetic energy.

Important Point

The correct answer is option 4) i.e. L = √(2EI)

CONCEPT:

• Angular momentum: The angular momentum of a rigid object is defined as the product of the moment of inertia and the angular velocity.
  • Angular momentum also obeys the law of conservation of momentum i.e. angular momentum before and after is conserved.

Angular momentum, L = I × ω 

Rotational kinetic energy: For a given fixed axis of rotation, the rotational kinetic energy is given by:

Where I is the moment of inertia, ω is the angular velocity.

CALCULATION:

Angular momentum, L = I × ω      ----(1) 

Roational kinetic energy

Substituting (2) in (1) we get

Concept:

The moment of inertia is the angular mass or rotational inertia. It is defined as a quantity that decides the amount of torque required for angular acceleration.  

The formula of Moment of Inertia is expressed as I = mr2 where m and r is mass and distance from the axis of rotation of the body. 

The SI unit of moment of inertia is kg m2.

The kinetic energy of a body in rotational motion is calculated using the formula:

where I is a moment of inertia and w is the angular velocity of the body in rotational motion.

Calculation: