Rotation

Question 1

What is the key difference between a point mass and a real object in terms of motion?

Answer 1

A point mass can only translate (move linearly) and does not rotate, while real objects can both translate and rotate due to their extended structure. Example: A box pushed at the top rotates in real life, not just stays at rest as a point mass model suggests.

Question 2

What is angular position

Answer 2

Angular position (θ) is a vector describing the orientation of a rotating object relative to a reference point. Its direction is determined by the right-hand rule: curl fingers in the rotation direction, and the thumb points along the axis (e.g., z-axis if rotation is in the xy-plane).

Question 3

How do you determine the direction of θ, ω, or α?

Answer 3

Use the right-hand rule: curl the fingers of your right hand in the direction of rotation (e.g., counterclockwise in the xy-plane), and your thumb points in the direction of the vector (e.g., +z-axis for counterclockwise rotation).

Question 4

What are the sign conventions for rotational motion in one dimension?

Answer 4

Counterclockwise rotation = Positive (+)
Clockwise rotation = Negative (-)
(Compare to translation: right = +, left = -)

Question 5

Define angular displacement and its units.

Answer 5

Angular displacement (θ(t)) is the change in angular position of a rotating object over time, measured in radians (rad).

Question 6

What is the formula for average angular velocity?

Answer 6

ω_av = (θ₂ - θ₁) / (t₂ - t₁)
Units: rad/s

Question 7

What is the formula for instantaneous angular velocity (angular frequency)?

Answer 7

ω(t) = dθ(t) / dt
Units: rad/s

Question 8

What is the formula for average angular acceleration?

Answer 8

α_av = (ω₂ - ω₁) / (t₂ - t₁)
Units: rad/s²

Question 9

What is the formula for instantaneous angular acceleration?

Answer 9

α(t) = dω(t) / dt
Units: rad/s²

Question 10

How does the direction of α affect ω?

Answer 10

If α is in the same direction as ω, angular speed increases (rotates faster).
If α is opposite to ω, angular speed decreases (slows down).

Question 11

How is angular velocity derived from angular acceleration?

Answer 11

ω(t) = ∫ α(t) dt + ω₀
where ω₀ is the initial angular velocity at t = 0 s.

Question 12

How is angular displacement derived from angular velocity?

Answer 12

θ(t) = ∫ ω(t) dt + θ₀
where θ₀ is the initial angular displacement at t = 0 s.

Question 13

List the kinematic equations for constant angular acceleration.

Answer 13

ω(t) = ω₀ + αt
θ(t) = θ₀ + ω₀t + (1/2)αt²
ω² = ω₀² + 2α(θ - θ₀)

Question 14

How are tangential velocity and angular velocity related?

Answer 14

v_t = rω
where v_t is tangential velocity (m/s), r is radius (m), and ω is angular velocity (rad/s).

Question 15

How are tangential acceleration and angular acceleration related?

Answer 15

a_t = rα
where a_t is tangential acceleration (m/s²), r is radius (m), and α is angular acceleration (rad/s²).

Question 16

What is the formula for centripetal acceleration in rotational motion?

Answer 16

a_c = v_t² / r = ω²r
where a_c is centripetal acceleration (m/s²), v_t is tangential velocity (m/s), and r is radius (m).

Question 17

How do you calculate the magnitude of total acceleration in circular motion?

Answer 17

a_total = √(a_t² + a_c²)
where a_t is tangential acceleration and a_c is centripetal acceleration.

Question 18

What is the kinetic energy of a point mass moving at angular velocity ω?

Answer 18

K = (1/2)mv_t² = (1/2)m(ωr)² = (1/2)(r²m)ω²
where m is mass (kg), v_t is tangential velocity (m/s), r is radius (m), and ω is angular velocity (rad/s).

Question 19

What is the formula for the kinetic energy of multiple point masses rotating at the same ω?

Answer 19

K = (1/2)(m₁r₁² + m₂r₂² + m₃r₃²)ω² = (1/2)(Σ r_j²m_j)ω²
where m_j is the mass of each point, r_j is its distance from the axis, and ω is angular velocity.

Question 20

What is the moment of inertia (I)?

Answer 20

The moment of inertia (I) is a measure of an object’s resistance to rotational acceleration about an axis, defined as:
I = Σ m_j r_j²
Units: kg·m²

Question 21

What is the formula for rotational kinetic energy?

Answer 21

K_rot = (1/2)Iω²
where I is moment of inertia (kg·m²) and ω is angular velocity (rad/s).

Question 22

What is the parallel-axis theorem?

Answer 22

I_parallel axis = I_center of mass + Md²
where I_parallel axis is the moment of inertia about a new axis, I_center of mass is the moment of inertia about the center of mass axis, M is mass (kg), and d is the distance between axes (m).

Question 23

What is torque (τ)?

Answer 23

Torque (τ) is the rotational equivalent of force, causing angular acceleration, defined as:
τ= r × F
Magnitude: τ = rF sin θ
Units: N·m

Question 24

How do you determine the direction of torque?

Answer 24

Use the right-hand rule: curl fingers from r to F (direction of rotation), and the thumb points in the direction of τ (e.g., +z for counterclockwise in the xy-plane).

Question 25

When does a non-zero force cause zero torque?

Answer 25

The force acts at the axis (r = 0). The force is parallel or antiparallel to the position vector (sin θ = 0).

Question 26

What is Newton’s Second Law for rotation?

Answer 26

Σ τ = Iα where Σ τ is the net torque (N·m), I is moment of inertia (kg·m²), and α is angular acceleration (rad/s²).

Question 27

What is the formula for work done by a constant torque?

Answer 27

W = τ(θ₂ - θ₁) where W is work (J), τ is torque (N·m), and θ₂ - θ₁ is angular displacement (rad).

Question 28

How do you calculate work for an angle-dependent torque?

Answer 28

W = ∫ τ(θ) dθ where τ(θ) is the torque as a function of angle, integrated over the angular displacement.

Question 29

What is the work-energy theorem for rotation?

Answer 29

W₁₂ = K₂ - K₁ = (1/2)Iω₂² - (1/2)Iω₁² where W₁₂ is the work done, K is rotational kinetic energy, I is moment of inertia, and ω₁, ω₂ are initial and final angular velocities.

Question 30

What is the total mechanical energy in a system with rotation?

Answer 30

E = K_trans + K_rot + U where K_trans = (1/2)mv² (translational kinetic energy), K_rot = (1/2)Iω² (rotational kinetic energy), and U = mgh (gravitational potential energy).

Question 31

What is the formula for power in rotational motion?

Answer 31

P = τω where P is power (W), τ is torque (N·m), and ω is angular velocity (rad/s).

Question 32

For a wind turbine rotating counterclockwise into the page, what are the directions of ω and α?

Answer 32

Using the right-hand rule, both ω (angular velocity) and α (angular acceleration) point out of the page (+z direction) for counterclockwise rotation into the page.

Question 33

Which has a larger moment of inertia: a bicycle wheel or a solid sphere (same mass, radius, ω)?

Answer 33

The bicycle wheel has a larger moment of inertia because its mass is distributed farther from the axis (I = MR² for a hoop vs. I = (2/5)MR² for a solid sphere).

Question 34

In rotation when in N = mg

Answer 34

when on a horisontal circle(on ground) N= mg. on vertical circle Fnet = N + w, where Fnet = ma_c.