Rotation Flashcards
What is the key difference between a point mass and a real object in terms of motion?
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.
What is angular position
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).
How do you determine the direction of θ, ω, or α?
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).
What are the sign conventions for rotational motion in one dimension?
Counterclockwise rotation = Positive (+)
Clockwise rotation = Negative (-)
(Compare to translation: right = +, left = -)
Define angular displacement and its units.
Angular displacement (θ(t)) is the change in angular position of a rotating object over time, measured in radians (rad).
What is the formula for average angular velocity?
ω_av = (θ₂ - θ₁) / (t₂ - t₁)
Units: rad/s
What is the formula for instantaneous angular velocity (angular frequency)?
ω(t) = dθ(t) / dt
Units: rad/s
What is the formula for average angular acceleration?
α_av = (ω₂ - ω₁) / (t₂ - t₁)
Units: rad/s²
What is the formula for instantaneous angular acceleration?
α(t) = dω(t) / dt
Units: rad/s²
How does the direction of α affect ω?
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).
How is angular velocity derived from angular acceleration?
ω(t) = ∫ α(t) dt + ω₀
where ω₀ is the initial angular velocity at t = 0 s.
How is angular displacement derived from angular velocity?
θ(t) = ∫ ω(t) dt + θ₀
where θ₀ is the initial angular displacement at t = 0 s.
List the kinematic equations for constant angular acceleration.
ω(t) = ω₀ + αt
θ(t) = θ₀ + ω₀t + (1/2)αt²
ω² = ω₀² + 2α(θ - θ₀)
How are tangential velocity and angular velocity related?
v_t = rω
where v_t is tangential velocity (m/s), r is radius (m), and ω is angular velocity (rad/s).
How are tangential acceleration and angular acceleration related?
a_t = rα
where a_t is tangential acceleration (m/s²), r is radius (m), and α is angular acceleration (rad/s²).
What is the formula for centripetal acceleration in rotational motion?
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).
How do you calculate the magnitude of total acceleration in circular motion?
a_total = √(a_t² + a_c²)
where a_t is tangential acceleration and a_c is centripetal acceleration.
What is the kinetic energy of a point mass moving at angular velocity ω?
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).
What is the formula for the kinetic energy of multiple point masses rotating at the same ω?
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.
What is the moment of inertia (I)?
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
What is the formula for rotational kinetic energy?
K_rot = (1/2)Iω²
where I is moment of inertia (kg·m²) and ω is angular velocity (rad/s).
What is the parallel-axis theorem?
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).
What is torque (τ)?
Torque (τ) is the rotational equivalent of force, causing angular acceleration, defined as:
τ= r × F
Magnitude: τ = rF sin θ
Units: N·m
How do you determine the direction of torque?
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).
When does a non-zero force cause zero torque?
The force acts at the axis (r = 0).
The force is parallel or antiparallel to the position vector (sin θ = 0).
What is Newton’s Second Law for rotation?
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).
What is the formula for work done by a constant torque?
W = τ(θ₂ - θ₁)
where W is work (J), τ is torque (N·m), and θ₂ - θ₁ is angular displacement (rad).
How do you calculate work for an angle-dependent torque?
W = ∫ τ(θ) dθ
where τ(θ) is the torque as a function of angle, integrated over the angular displacement.
What is the work-energy theorem for rotation?
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.
What is the total mechanical energy in a system with rotation?
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).
What is the formula for power in rotational motion?
P = τω
where P is power (W), τ is torque (N·m), and ω is angular velocity (rad/s).
For a wind turbine rotating counterclockwise into the page, what are the directions of ω and α?
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.
Which has a larger moment of inertia: a bicycle wheel or a solid sphere (same mass, radius, ω)?
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).
In rotation when in N = mg
when on a horisontal circle(on ground) N= mg. on vertical circle Fnet = N + w, where Fnet = ma_c.
