Dynamics of rotational motions – problems and solutions

회전 운동의 역학 – 문제점과 해결책

1. 도르래 관성 모멘트 I = 2/5 MR² 질량이 2kg입니다. 만약 힘의 순간 도르래에 걸리는 힘이 4 Nm일 때, 도르래의 선형 가속도는 얼마입니까? 중력 가속도는 g = 10 m²/s²입니다.^-2.

알려진 : 회전 운동의 역학 – 문제점 및 해결책 1

도르래의 관성 모멘트(I) = 2/5 MR²

질량 풀리의 질량(M) = 2kg

모멘트(τ) = 4 Nm

중력에 의한 가속 (g) = 10ms^-2

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

구함: The linear acceleration (a)

솔루션 :

The moment of inertia of pulley (I) :

I = 2/5 MR²= 2/5 (2)(0.2)² = 2/5 (2)(0.04) = 2/5 (0.08) = 0.16 / 5 = 0.032 kg m²

The 각가속도 (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.

알려진 바에 따르면:

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

중력 가속도(g) = 10 m/s²^-2

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

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

구함: The moment of inertia of pulley

참조 관성 모멘트 방정식

솔루션 :

The moment of force (τ) :

τ = FR = 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 m²

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

알려진 바에 따르면:

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

중력 가속도(g) = 10 m/s²^-2

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

힘(F) = 4 N

구함: The moment of inertia of the pulley

솔루션 :

The moment of force (τ) :

τ = FR = (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 m²

What is the rotational equivalent of mass in linear motion?
- 답변 : 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?
- 답변 : 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?
- 답변 : The relationship is given by Newton’s second law for rotation: $τ = I α$ 어디로 $τ$ is the torque, $I$ 는 관성 모멘트입니다. $α$ 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?
- 답변 : 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?
- 답변 : Linear momentum (given by $p = m v$ 어디로 $m$ 질량이고 $v$ is velocity) is the product of mass and linear velocity, whereas angular momentum (given by $L = I ω$ 어디로 $I$ 관성 모멘트이며 $ω$ is the angular velocity) is the product of the moment of inertia and angular velocity.

What is meant by the conservation of angular momentum?
- 답변 : 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?
- 답변 : 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?
- 답변 : The rotational kinetic energy (given by 1/2 $I ω^{2}$ ) is analogous to linear kinetic energy (given by 1/2 $m v^{2}$ ). 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?
- 답변 : 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?

답변 : The moment of inertia of a solid disk is given by 1/2 $m r^{2}$ 어디로 $m$ 질량이며 $r$ is the radius. If the radius doubles and the mass remains constant, the moment of inertia will increase by a factor of four.