Worksheet with Solutions: System of Particles and Rotational Motion | Physics Class 11 - NEET PDF Download

Ans: b) It can lie outside the body depending on mass distribution.
The center of mass can lie outside the physical boundaries of the system, particularly when the mass distribution is uneven. Thus, option b is correct while the others misrepresent the conditions of the center of mass.

Ans: (c) ML²/12
The moment of inertia of a thin rod about its center is calculated as ML²/12, where M is the mass and L is the length of the rod. Therefore, option C is correct while the others are incorrect values.

Ans: (b) Angular velocity
In pure rotational motion, angular velocity remains constant for all points in the rotating body, making option b the correct choice. The other options do not apply to all particles in rotation.

Ans: (b) τ = Iα
The equation τ = Iα relates torque (τ) to angular acceleration (α) through moment of inertia (I), making option B the correct answer. The other equations pertain to linear motion or momentum.

Ans: (d) They have the same angular velocity.
In a rigid body rotating about a fixed axis, all particles experience the same angular velocity, which is why option d is the correct answer. The other options misrepresent the characteristics of rigid body rotation.

Ans: center of mass
The center of mass is a crucial concept in physics, representing the average position of mass in a system of particles.

Ans: moment of inertia
The moment of inertia quantifies how difficult it is to change the rotational motion of an object, depending on its mass distribution.

Ans: angular momentum
Angular momentum reflects the rotational motion of an object and is conserved in isolated systems.

Ans: torque
Torque is the measure of the force that can cause an object to rotate around an axis, essential in understanding rotational dynamics.

Ans: conserved
In the absence of external torque, the conservation of angular momentum principle states that the total angular momentum remains constant.

Ans: True
A rigid body can indeed exhibit both translational motion (movement from one location to another) and rotational motion (spinning around an axis) at the same time.

Ans: True
The center of mass for a uniform object, such as a sphere or cube, is positioned at its geometric center, due to uniform mass distribution.

Ans: False
In rotational motion, all particles of a rigid body have the same angular velocity, although they can have different linear velocities depending on their distance from the axis of rotation.

Ans: False
Torque is a vector quantity because it has both magnitude and direction, affecting the rotational motion of an object.

Ans: True
The moment of inertia is influenced by both the mass of the object and how that mass is distributed concerning the axis of rotation, affecting its resistance to rotational motion.

Sol: 

Ans: A rigid body is an object that does not change its shape or size when forces are applied to it. This means that the distance between any two points in the body remains constant, regardless of the forces acting upon it.

Ans: The center of mass is a specific point within an object where all its mass can be considered to be concentrated. It acts as the balance point of the object. When an object is supported at this point, it remains balanced and does not tip over.

Ans: When a rigid body, such as a cylinder, rolls down an incline, it exhibits a combination of two types of motion: it rolls (turns) and slides (moves downward). Every part of the cylinder moves together, but the point in contact with the ground remains stationary for a brief moment.

Ans: Torque is a measure of how much a force can make something rotate. Think of it like the twist you give to a door handle. The harder you push or pull at the edge of the door, the more it will turn.

Ans: Skaters spin faster when they pull their arms in because they reduce their moment of inertia. This action concentrates their mass closer to the centre of their body, which allows them to spin more quickly without requiring additional energy.

Ans: The centre of mass (CM) of a system of particles is a crucial concept in physics that represents a point where the total mass of a system can be considered to be concentrated. This point is determined mathematically by the weighted average of the positions of all particles in the system, taking each particle's mass into account. The position of the centre of mass is given by the formula:

 Definition: For a system of n particles with masses m₁, m₂, ..., m_n located at positions r₁, r₂, ..., r_n, the centre of mass R is defined as: R = (m₁r₁ + m₂r₂ + ... + m_nr_n) / (m₁ + m₂ + ... + m_n). Significance: The CM allows us to simplify complex systems. Instead of analysing the motion of every particle, we can treat the entire system as if all its mass were concentrated at the CM. This simplification is particularly useful in mechanics, where we apply Newton's laws to predict motion. 

Motion of the Centre of Mass: The motion of the centre of mass is governed by external forces acting on the system. If no external forces act, the CM will move with constant velocity, demonstrating the principle of conservation of momentum.

• Example - Two-Particle System: Consider two particles of equal mass m located at points x₁ and x₂. The CM lies exactly halfway between them at (x₁ + x₂) / 2. This example illustrates that the CM is influenced directly by the distribution of mass.
• Example - Extended Bodies: When analysing an object like a uniform rod, the CM can be found at its geometrical centre. This knowledge is critical in applications such as balancing objects, designing structures, and understanding stability in physics.

Ans: In the study of rotational motion about a fixed axis, three fundamental concepts are interrelated: angular velocity (ω), linear velocity (v), and moment of inertia (I). Understanding these relationships is essential for analysing the motion of rotating bodies.

 Angular Velocity (ω): Angular velocity is defined as the rate of change of angular displacement with respect to time. It is a vector quantity that describes how fast an object rotates around a fixed axis. The formula is ω = dθ/dt, where dθ is the angular displacement. 

Linear Velocity (v): Linear velocity refers to the speed of a point on a rotating object. It is given by the formula v = ωr, where r is the distance from the axis of rotation to the point in question. This relationship indicates that points farther from the axis move faster than those closer to it. 

Moment of Inertia (I): The moment of inertia quantifies how mass is distributed relative to the axis of rotation. It is calculated using I = Σmr², where m is the mass of the particles and r is their distance from the axis. A larger I indicates that more effort (torque) is required to change the rotational speed of the body.

• Example - Solid Cylinder: Consider a solid cylinder rolling down an incline. The linear acceleration of the centre of mass can be derived from the torque due to gravity acting on the cylinder, considering I. This helps in understanding how rotational motion converts into linear motion.
• Example - Flywheel: In applications like flywheels, which store rotational energy, understanding the relationship between ω and I is crucial. For a flywheel with a large moment of inertia, a small torque can result in a large change in angular velocity, demonstrating its ability to maintain steady speeds despite fluctuations in the applied torque.

Ans: Option (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Explanation:

• A sliding disc has only translational kinetic energy: (1/2)mv²

• A rolling disc has both translational and rotational kinetic energy:
  → Translational: (1/2)mv²
  → Rotational: (1/2)Iω² = (1/2) × (1/2)mr² × (v/r)² = (1/4)mv²
  → Total = (1/2 + 1/4)mv² = (3/4)mv²
  So, rolling has more energy, thus confirming that both A and R are correct, with R explaining A.

Ans: Option (c) Assertion (A) is true but reason (R) is false.
Explanation:

• The motion of the center of mass depends only on external forces.

• Internal forces always cancel in pairs (Newton's 3rd law) and do not affect center of mass motion.

• Thus, assertion (A) is true, while reason (R) is false.

Ans: Option (a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
Explanation:

• Acceleration while rolling = (g sinθ) / (1 + (k²/R²))

• For a solid sphere, k² = (2/5)R² → lower denominator → more acceleration

• For a hollow sphere, k² = R² → larger denominator → less acceleration

• Thus, the hollow sphere rolls slower and reaches the bottom last. Both A and R are correct, with R explaining A.

Ans: Angular Momentum (L):
For circular motion, the formula for angular momentum is:
L = m × r² × ω
Substituting the given values:
L = 0.2 kg × (0.5 m)² × 10 rad/s
L = 0.2 kg × 0.25 m² × 10 rad/s = 0.5 kg·m²/s Torque (τ):
In uniform rotation, the angular speed (ω) remains constant. Therefore, the angular acceleration (α) is:
α = 0
Using the formula for torque:
τ = I × α = 0
Thus, the torque required to maintain uniform rotation is 0 N·m.

Ans: A flywheel with a moment of inertia of 0.2 kg·m² is rotating at 60 rad/s. A torque of -0.4 N·m is applied. To find the time taken to stop, we can use the formula: Torque = I × α Where:

• τ (Torque) = -0.4 N·m
• I (Moment of Inertia) = 0.2 kg·m²
• α (Angular Acceleration) = τ / I

Calculating α: α = -0.4 / 0.2 = -2 rad/s² Now, we can use the equation: ω = ω₀ + αt Substituting the values: 0 = 60 + (-2)t Thus, we have: -60 = -2t Solving for t: 
t = 30 s

Ans: Moment of Inertia (about the end) is calculated as follows: I = (1/3)ML² = (1/3) × 2 kg × (3 m)² = 6 kg·m²
The Initial Potential Energy is given by: PE = Mg(L/2) = 2 kg × 10 m/s² × 1.5 m = 30 J
At the horizontal position, all potential energy converts into rotational kinetic energy: (1/2)Iω² = 30 J Thus, 0.5 × 6 kg·m² × ω² = 30 J ⇒ ω² = 10 ⇒ ω = √10 ≈ 3.16 rad/s