گردشی حرکت - مسائل اور حل

گردشی حرکت - مسائل اور حل

Torque

1. A beam 140 cm in length. There are three forces acts on the beam, F₁ = 20 ن، ایف₂ = 10 N, and F₃ = 40 N with direction and position as shown in the figure below. What is the ٹارک causes the beam rotates about the center of mass of the beam?

جانا جاتا ہے:

The center of mass located at the center of the beam.

Length of beam (l) = 140 cm = 1.4 meters

فورس 1 (F₁) = 20 N, the lever arm 1 (l₁) = 70 cm = 0.7 meters

فورس 2 (F₂) = 10 N, the lever arm 2 (l₂) = 100 cm – 70 cm = 30 cm = 0.3 meters

فورس 3 (F₃) = 40 N, the lever arm 3 (l₃) = 70 cm = 0.7 meters

مطلوب: The magnitude of torque

حل:

The torque 1 rotates beam clockwise, so assigned a negative sign to the torque 1.

τ₁ = ایف₁ l₁ = (20 N) (0.7 m) = -14 N m

The torque 2 rotates beam counterclockwise, so assigned a positive sign to the torque 2.

τ₂ = ایف₂ l₂ = (10 N) (0.3 m) = 3 N m

The torque 3 rotates beam clockwise, so assigned a positive sign to the torque 3.

τ₃ = ایف₃ l3 = (40 N)(0.7 m) = -28 N m

The net torque :

Στ = -14 Nm + 3 Nm – 28 Nm = – 42 Nm + 3 Nm = -39 Nm

The magnitude of the torque is 39 N m. The direction of rotation of the beam clockwise, so assigned a negative sign.

2. What is the net torque acts on the beam The axis of rotation at point D. (sin 53^o = 0.8)

جانا جاتا ہے:

The axis of rotation at point D

F₁ = 10 N and l₁ = ر₁ sin θ = (40 cm)(sin 53^o) = (0.4 m)(0.8) = 0.32 meters

F₂ = 10√2 N and l₂ = ر₂ sin θ = (20 cm)(sin 45^o) = (0.2 m)(0.5√2) = 0.1√2 meters

F₃ = 20 N and l₃ = ر₁ sin θ = (10 cm)(sin 90^o) = (0.1 m)(1) = 0.1 meters

مطلوب: The net torque

حل:

τ₁ = ایف₁ l₁ = (10 N)(0.32 m) = 3.2 Nm

(The torque 1 rotates beam counterclockwise so we assign positive sign to the torque 1)

τ₂ = ایف₂ l₂ = (10√2 N)( 0.1√2 m) = -2 Nm

(The torque 2 rotates beam clockwise so we assign negative sign to the torque 2)

τ₃ = ایف₂ l₂ = (20 N)(0.1 m) = 2 Nm

(The torque 3 rotates beam counterclockwise so we assign positive sign to the torque 3)

The net torque :

Στ = τ₁ - τ₁ + τ₃

Στ = 3.2 Nm – 2 Nm + 2 Nm

Στ = 3.2 Nm

3. What is the net torque if the axis of rotation at point D. (sin 53^o = 0.8)

جانا جاتا ہے:

The axis of rotation at point D.

فاصلہ between F₁ and the axis of rotation (r_AD) = 40 سینٹی میٹر = 0.4 میٹر

Distance between F₂ and the axis of rotation (r_BD) = 20 سینٹی میٹر = 0.2 میٹر

Distance between F₃ and the axis of rotation (r_CD) = 10 سینٹی میٹر = 0.1 میٹر

F₁ = 10 نیوٹن

F₂ = 10√2 نیوٹن

F₃ = 20 نیوٹن

گناہ 53^o 0.8 =

مطلوب: The net torque

حل:

The moment of the force 1

Στ₁ = (ایف₁)(ر_AD گناہ 53^o) = (10 N)(0.4 m)(0.8) = 3.2 N.m

(The torque 1 rotates beam counterclockwise so we assign positive sign to the torque 1)

The moment of the force 2

Στ₂ = (ایف₂)(ر_BD گناہ 45^o) = (10√2 N)(0.2 m)(0.5√2) = -2 N.m

(The torque 2 rotates beam clockwise so we assign negative sign to the torque 2)

The moment of the force 3

Στ₃ = (ایف₃)(ر_CD گناہ 90^o) = (20 N)(0.1 m)(1) = 2 N.m

(The torque 2 rotates beam counterclockwise so we assign positive sign to the torque 3)

The net torque :

Στ = Στ₁ + Στ₂ + Στ₃

Στ = 3.2 – 2 + 2

Στ = 3.2 Newton meter

جڑتا کا لمحہ

4. Length of wire = 12 m, l₁ = 4 m. Ignore wire’s mass. What is the جڑتا کا لمحہ نظام کی.

جانا جاتا ہے:

ماس of A (m_A) = 0.2 کلو

Mass of B (m_B) = 0.6 کلو

Distance between A and the axis of rotation (r_A) = 4 میٹر

Distance between B and the axis of rotation (r_B) = 12 – 4 = 8 meters

مطلوب: The moment of inertia of the system

حل:

The moment of inertia of A

I_A = (م_A)(ر_A²) = (0.2)(4)² = (0.2)(16) = 3.2 کلو میٹر²

The moment of inertia of B

I_B = (م_B)(ر_B²) = (0.6)(8)² = (0.6)(64) = 38.4 کلو میٹر²

The moment of inertia of the system :

میں = I_A + I_B = 3.2 + 38.4 = 41.6 kg m²

Rotational dynamics

5. A 6-N force is applied to a cord wrapped around a pulley of mass M = 5 kg and radius R = 20 cm. What is the angular acceleration of the pulley. The pulley is a uniform solid cylinder.

جانا جاتا ہے:

فورس (F) = 6 نیوٹن

Mass (M) = 5 kg

Radius (R) = 20 cm = 20/100 m = 0.2 m

مطلوب: کونیی ایکسلریشن (α)

حل:

The moment of the force :

τ = F R = (6 Newton)(0.2 meters) = 1.2 N m

The moment of inertia for solid cylinder :

I = 1/2 MR²

I = 1/2 (5 kg)(0.2 m)²

I = 1/2 (5 kg)(0.04 m²)

I = 1/2 (0.2)

I = 0.1 کلوگرام میٹر².

The angular acceleration :

τ = میں α

α = τ / I = 1.2 / 0.1 = 12 rad s^-2

6. A block of mass = 4 kg hanging from a cord wrapped around a pulley of mass = 8 kg and radius R = 10 cm. Acceleration due to gravity is 10 ms^-2 . What is the linear acceleration of the block? The pulley is a uniform solid cylinder.

جانا جاتا ہے:

Mass of pulley (m) = 8 kg

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

بلاک کا ماس (m) = 4 کلوگرام

کشش ثقل کی وجہ سے سرعت (g) = 10 m/s²

وزن (w) = mg = (4 kg) (10 m/s²) = 40 کلو میٹر فی سیکنڈ² = 40 نیوٹن

مطلوب: The free fall acceleration of the block

حل:

The moment of inertia of the solid cylinder :

I = 1/2 MR² = 1/2 (8 kg)(0.1 m)² = (4 kg)(0.01 m²) = 0.04 کلوگرام میٹر²

The moment of the force :

τ = F r = (40 N)(0.1 m) = 4 Nm

The angular acceleration :

Στ = میں α

4 = 0.04 α

α = 4 / 0.04 = 100

The linear acceleration :

a = r α = (0.1)(100) = 10 m/s²

7. A block with mass of m hanging from a cord wrapped around a pulley. If the مفت زوال acceleration of the block is a m/s², what is the moment of inertia of the pulley..

جانا جاتا ہے:

weight = w = m g

Lever arm = R

The angular acceleration = α

The free fall acceleration of the block = a ms^-2

مطلوب: The moment of inertia of the pulley (I)

حل:

The connection between the linear acceleration and the angular acceleration :

a = R α

α = a / R

The moment of inertia :

τ = میں α

I = τ : α = τ : a / R = τ (R / a) = τ R a^-1

The angular momentum

8. A 0.2-gram particle moves in a circle at a constant speed of 10 m/s. The radius of the circle is 3 cm. What is the angular momentum of the particle?

جانا جاتا ہے:

Mass of particle (m) = 0.2 gram = 2 x 10^-4 kg

Angular speed (ω) = 10 rad s^-1

Radius (r) = 3 cm = 3 x 10^-2 میٹر

مطلوب: The angular momentum of the particle

حل:

The equation of the angular momentum :

L = میں ω

I = the angular momentum, I = the moment of inertia, ω = the angular speed

The moment of inertia (for particle) :

میں = مسٹر² = (2 x 10۔^-4 )(3 x 10^-2)² = (2 x 10۔^-4 )(9 x 10^-4) = 18 x 10^-8

کونیی رفتار:

L = I ω = (18 x 10^-8)(10 rad s^-1) = 18 x 10^-7 کلو میٹر² s^-1

1. What is rotational motion?
   • کا جواب: Rotational motion refers to the movement of an object around a fixed axis. It’s the kind of motion in which every point of the object moves in a circle about the axis.
2. How does linear velocity relate to angular velocity in rotational motion?
   • کا جواب: Linear velocity ($v$) of a point in a rotating object is directly proportional to its distance ($r$) from the axis of rotation and the angular velocity ($\omega$) of the object. The relation is given by $v=r\cdot \omega$.
3. What is the moment of inertia, and how does it relate to rotational motion?
   • کا جواب: Moment of inertia is the rotational analog of mass in linear motion. It measures an object’s resistance to changes in its rotational state. The moment of inertia depends on both the mass of an object and its distribution relative to the axis of rotation.
4. How does Newton’s first law of motion apply to rotational motion?
   • کا جواب: Just as an object in linear motion remains in motion unless acted upon by an external force, an object in rotational motion will remain in that state unless acted upon by an external torque.
5. What is the significance of the radius of gyration?
   • کا جواب: The radius of gyration provides a measure of the distribution of an object’s mass away from its axis of rotation. It essentially describes how far from the axis all of the object’s mass would need to be concentrated to have the same moment of inertia as the original distribution.
6. What is angular momentum and how is it conserved?
   • کا جواب: Angular momentum is the rotational equivalent of linear momentum. It is the product of an object’s moment of inertia and its angular velocity. In a closed system, the total angular momentum remains constant unless acted upon by an external torque, highlighting the conservation of angular momentum.
7. How does torque influence rotational motion?
   • کا جواب: Torque is the rotational equivalent of force. It causes changes in the rotational motion of an object. The relation is given by Newton’s second law for rotation: $\tau =I\cdot \alpha$، کہاں $\tau$ is the torque, $I$ is the moment of inertia, and $\alpha$ is the angular acceleration.
8. How does the center of mass differ from the center of rotation?
   • کا جواب: While they can coincide, the center of mass is the point where the entire mass of an object can be assumed to be concentrated for the purpose of calculations in linear motion, whereas the center of rotation is the point (or axis) about which an object rotates.
9. What is the role of the centripetal force in rotational motion?
   • کا جواب: Centripetal force is the net force acting on an object moving in a circular path, directed towards the center of rotation. It’s responsible for keeping an object in its curved path and preventing it from moving in a straight line due to inertia.

10. How is rotational kinetic energy related to the moment of inertia and angular velocity?

    • کا جواب: Rotational kinetic energy is the energy due to the rotation of an object about an axis. It is given by the formula: KE_{rotational​}=1/2 ​I ω²، کہاں $I$ is the moment of inertia and $\omega$ is the angular velocity.