Unit 6: Energy and Momentum of Rotating Systems

Question 1

What is the formula for rotational kinetic energy?

Answer 1

Rotational kinetic energy (KE_rot) = 1/2 I ω², where I is the moment of inertia and ω is the angular velocity.

Question 2

True or False: The moment of inertia depends on the mass distribution of an object.

Answer 2

True

Question 3

Fill in the blank: The unit of moment of inertia in the SI system is __________.

Answer 3

kg·m²

Question 4

What is the relationship between torque (τ), moment of inertia (I), and angular acceleration (α)?

Answer 4

τ = I α

Question 5

What is the principle of conservation of angular momentum?

Answer 5

The total angular momentum of a closed system remains constant if no external torques act on it.

Question 6

If a rotating object speeds up, what happens to its angular momentum?

Answer 6

It increases, assuming no external torques are acting.

Question 7

What is the equation for angular momentum (L)?

Answer 7

L = I ω

Question 8

True or False: Angular momentum is a vector quantity.

Answer 8

True

Question 9

Define the term ‘moment of inertia’.

Answer 9

Moment of inertia is a measure of an object’s resistance to changes in its rotation.

Question 10

What is the formula for the moment of inertia of a solid cylinder about its central axis?

Answer 10

I = 1/2 m r², where m is mass and r is radius.

Question 11

What is the relationship between linear velocity (v) and angular velocity (ω)?

Answer 11

v = r ω, where r is the radius.

Question 12

Fill in the blank: The work done by a torque is equal to the change in __________.

Answer 12

rotational kinetic energy

Question 13

What is the formula for work done by torque?

Answer 13

W = τ θ, where θ is the angular displacement in radians.

Question 14

True or False: The angular displacement must be in radians when using the work done by torque equation.

Answer 14

True

Question 15

What is the unit of torque in the SI system?

Answer 15

Newton-meter (N·m)

Question 16

What is the relationship between angular momentum (L) and torque (τ)?

Answer 16

τ = dL/dt, meaning torque is the rate of change of angular momentum.

Question 17

Fill in the blank: A rotating object that is not subject to external torque will conserve its __________.

Answer 17

angular momentum

Question 18

What is the formula for the moment of inertia of a thin spherical shell?

Answer 18

I = 2/3 m r²

Question 19

True or False: Increasing the radius of a rotating object increases its moment of inertia.

Answer 19

True

Question 20

What is the effect of doubling the angular velocity on the rotational kinetic energy?

Answer 20

It increases the rotational kinetic energy by a factor of four, since KE_rot ∝ ω².

Question 21

What is the equation for angular acceleration?

Answer 21

α = Δω/Δt, where Δω is the change in angular velocity and Δt is the change in time.

Question 22

Fill in the blank: In a system where no external torque is applied, the angular momentum before an event is equal to the angular momentum __________ the event.

Answer 22

after

Question 23

What is the formula for the moment of inertia of a solid disk about its central axis?

Answer 23

I = 1/2 m r²

Question 24

True or False: The angular momentum of a particle moving in a circular path is constant if the speed is constant.

Answer 24

True

Question 25

What is the formula for the conservation of angular momentum when an ice skater pulls in their arms?

Answer 25

L_initial = L_final, or I_initial ω_initial = I_final ω_final.

Question 26

Fill in the blank: The angular velocity of an object is measured in __________.

Answer 26

radians per second (rad/s)

Question 27

What is the definition of torque?

Answer 27

Torque is a measure of the force that produces or changes rotation.

Question 28

What is the relationship between linear acceleration (a) and angular acceleration (α)?

Answer 28

a = r α, where r is the radius.

Question 29

Fill in the blank: The total mechanical energy of a rotating system is the sum of its __________ and __________.

Answer 29

kinetic energy, potential energy

Question 30

What is the equation for the potential energy of a rotating object at height h?

Answer 30

PE = mgh, where m is mass, g is gravitational acceleration, and h is height.

Question 31

True or False: The direction of torque is determined by the right-hand rule.

Answer 31

True

Question 32

What is the angular momentum of a particle moving in circular motion?

Answer 32

L = mvr, where m is mass, v is linear velocity, and r is radius.

Question 33

What is the formula for the moment of inertia of a rectangular plate about an axis through its center?

Answer 33

I = (1/12) m (a² + b²), where a and b are the dimensions of the plate.

Question 34

Fill in the blank: A flywheel is a device that stores __________ energy in the form of rotation.

Answer 34

kinetic

Question 35

What is the effect of increasing the moment of inertia on the angular acceleration of an object when torque is constant?

Answer 35

Angular acceleration decreases as moment of inertia increases.

Question 36

True or False: The rotational analog of Newton's second law states that τ = F r.

Answer 36

False; it should be τ = r F sin(θ).

Question 37

What is the relationship between the radius and moment of inertia for a point mass?

Answer 37

I = m r², where m is the mass and r is the distance from the axis of rotation.

Question 38

Fill in the blank: The rotational equivalent of mass is __________.

Answer 38

moment of inertia

Question 39

What is the formula for the total kinetic energy of a system of particles in rotation?

Answer 39

KE_total = Σ(1/2 I_i ω_i²) for each particle i.

Question 40

True or False: An object in uniform circular motion has a constant angular velocity.

Answer 40

True

Question 41

What is the relationship between angular displacement and angular velocity?

Answer 41

ω = dθ/dt, where dθ is the angular displacement and dt is the change in time.

Question 42

Fill in the blank: The SI unit of angular momentum is __________.

Answer 42

kg·m²/s

Question 43

What is the formula for the moment of inertia of a hollow cylinder about its central axis?

Answer 43

I = m r²

Question 44

What does the term 'rotational equilibrium' mean?

Answer 44

Rotational equilibrium occurs when the sum of the torques acting on an object is zero.

Question 45

Fill in the blank: The angular momentum of a system is conserved in the absence of __________.

Answer 45

external torques

Question 46

What is the formula for the torque applied by a force at an angle θ?

Answer 46

τ = r F sin(θ)

Question 47

True or False: The angular velocity of an object can change even if the speed remains constant.

Answer 47

True; it can change direction.

Question 48

What is the relationship between the moment of inertia and the radius for a rotating object?

Answer 48

The moment of inertia increases with the square of the radius.

Question 49

Fill in the blank: The principle that describes a rotating object's tendency to maintain its state of motion is called __________.

Answer 49

rotational inertia

Question 50

What is the angular momentum of a system of particles?

Answer 50

L = Σ(m_i r_i v_i), where m_i is the mass, r_i is the position vector, and v_i is the velocity of each particle.

Question 51

What is the significance of the right-hand rule in rotational motion?

Answer 51

It is used to determine the direction of angular velocity and torque.

Question 52

Fill in the blank: The kinetic energy of a rotating object increases with the square of its __________.

Answer 52

angular velocity

Question 53

What is the equation for the conservation of mechanical energy in a rotating system?

Answer 53

KE_initial + PE_initial = KE_final + PE_final