Physics Chapter on Rolling Motion

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Questions and Answers

What is the expression for total angular momentum when considering rolling motion?

  • $L = ΣI + Σr_i m_i^2 v_i$
  • $L = ΣI + Σr_i m_i v_i$ (correct)
  • $L = ΣI + Σm_iv_i$
  • $L = ΣI + Σr_i^2 m_i v_i$

Under which condition does a body exhibit pure rolling motion?

  • When the body is sliding down the incline
  • When the velocity of the contact point is zero (correct)
  • When the velocity of the contact point exceeds the body velocity
  • When there is no friction acting on the body

What is the minimum coefficient of friction required for a body to roll down an incline without slipping?

  • $ an heta$
  • $ rac{g heta}{1}$
  • $ rac{1}{g heta}$
  • $ rac{1}{ an heta}$ (correct)

What happens when the coefficient of friction is less than the minimum required for pure rolling?

<p>The body slips downwards with translational acceleration only (D)</p> Signup and view all the answers

Which expression correctly represents the acceleration of a body rolling down an incline when friction is present and sufficient?

<p>$a = rac{g an heta}{1 + rac{I}{mR^2}}$ (C)</p> Signup and view all the answers

What is the primary factor that influences the momentum of an object?

<p>Mass of the object (C)</p> Signup and view all the answers

Which of the following correctly represents the formula for kinetic energy?

<p>$KE = \frac{1}{2}mv^2$ (B)</p> Signup and view all the answers

Which principle states that the total mechanical energy in a closed system remains constant?

<p>Conservation of energy (A)</p> Signup and view all the answers

In a perfectly elastic collision, which of the following is conserved?

<p>Both momentum and kinetic energy (C)</p> Signup and view all the answers

What is the unit of measurement for power in the International System of Units (SI)?

<p>Watt (C)</p> Signup and view all the answers

When an object is in free fall, ignoring air resistance, which of the following changes over time?

<p>Velocity of the object (B)</p> Signup and view all the answers

Which of the following is true about gravitational potential energy?

<p>It is directly proportional to height and mass. (C)</p> Signup and view all the answers

In the context of waves, which property determines the pitch of a sound?

<p>Frequency of the wave (B)</p> Signup and view all the answers

What type of energy is stored in objects due to their position or state?

<p>Potential energy (C)</p> Signup and view all the answers

Which of the following best describes the relationship between work and energy?

<p>Work transfers energy. (C)</p> Signup and view all the answers

What is the correct formula for the $x_{cm}$ of a semicircular ring?

<p>$x_{cm} = 0$ (A)</p> Signup and view all the answers

When differentiating the position of the center of mass with respect to time, which quantity represents its velocity?

<p>The derivative of the center of mass coordinates (C)</p> Signup and view all the answers

According to Newton's second law, which relationship is correctly applied to the center of mass?

<p>$F = ma_{cm}$ (C)</p> Signup and view all the answers

What principle states that the total momentum of a closed system remains constant?

<p>Conservation of linear momentum (B)</p> Signup and view all the answers

The direction of the cross product of two vectors can be determined using which method?

<p>Right-hand rule (B)</p> Signup and view all the answers

For a rectangular plate, how is the center of mass typically positioned?

<p>At the geometric center of the plate (C)</p> Signup and view all the answers

What is required to calculate the magnitude of the cross product of two vectors?

<p>The angle between the vectors and their lengths (C)</p> Signup and view all the answers

In a system where two particles collide, which statement about the center of mass velocity is true?

<p>Velocity of the center of mass is independent of external forces (A)</p> Signup and view all the answers

What happens to the center of mass of a system when two particles of equal mass collide elastically?

<p>It remains stationary (C)</p> Signup and view all the answers

If an external force acts on a system of particles, how does it affect the center of mass?

<p>The acceleration of the center of mass is proportional to the net external force (C)</p> Signup and view all the answers

Which of the following best describes a rigid body?

<p>A body that maintains a constant distance between all particles (A)</p> Signup and view all the answers

In rotational motion, how does each particle of a rigid body move?

<p>In a circle around the axis of rotation (D)</p> Signup and view all the answers

Which equation calculates the centre of mass (CM) of two particles with masses $m_1$ and $m_2$?

<p>$ ext{CM} = rac{m_1 ext{r}<em>{1} + m_2 ext{r}</em>{2}}{m_1 + m_2}$ (D)</p> Signup and view all the answers

For which case is the centre of mass representation most complex?

<p>For a continuous distribution of mass in a rigid body (D)</p> Signup and view all the answers

What does the variable $M$ represent in the centre of mass equations?

<p>Total mass of the system (C)</p> Signup and view all the answers

How does precession change the motion of a spinning body?

<p>The axis of rotation moves around a vertical axis (B)</p> Signup and view all the answers

Which of the following statements is true regarding translational motion?

<p>All particles of a body move with the same velocity at any instant (B)</p> Signup and view all the answers

What is the formula for the $x_{CM}$ position of a rigid body in a continuous mass distribution?

<p>$x_{CM} = rac{1}{M} imes ext{Integrand of } x ext{ dm}$ (A)</p> Signup and view all the answers

Which option correctly represents the relationship between rotational and translational motion?

<p>Rotational motion involves movement in circles around an axis (C)</p> Signup and view all the answers

What is the primary condition for a body to be in rotational equilibrium?

<p>The net torque acting on the body must be zero. (B)</p> Signup and view all the answers

Which equation accurately relates linear velocity to angular velocity for a particle at a distance $r$ from the center of rotation?

<p>$v = r imes rac{d heta}{dt}$ (A)</p> Signup and view all the answers

How is angular momentum defined in relation to linear momentum?

<p>$ ext{L} = ext{r} imes ext{p}$ where $ ext{p} = m ext{v}$ (D)</p> Signup and view all the answers

Which physical quantity is analogous to mass in rotational motion?

<p>Moment of inertia (C)</p> Signup and view all the answers

When a body is in translational equilibrium, which of the following must be true?

<p>The sum of all external forces acting on it is zero. (A)</p> Signup and view all the answers

Which statement best describes torque in relation to force?

<p>Torque is the rotational analogue of force and depends on distance and angle. (A)</p> Signup and view all the answers

What does the principle of moments state about forces acting on a body?

<p>If the sum of moments about a fixed point is zero, the body is in rotational equilibrium. (C)</p> Signup and view all the answers

Which of the following correctly states the relationship between the time rate of angular momentum and torque?

<p>$ rac{dL}{dt} = au$ (B)</p> Signup and view all the answers

What is the formula for the total kinetic energy of a rigid body in rotational motion?

<p>$KE = rac{1}{2} imes ext{I} imes ext{ω}^2$ (C)</p> Signup and view all the answers

Flashcards

Rigid body

A body with a fixed shape; distances between particles don't change.

Translational motion

All particles move with the same velocity. Moves as a block.

Rotational motion

A body rotates around a fixed axis. Each particle moves in a circle.

Precession

The axis of rotation changes, making a cone.

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Center of Mass (CM)

The point where the total mass is concentrated.

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CM of two particles

Position found using weighted average of positions based on mass.

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CM of n-particles

Weighted average of all positions based on mass.

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CM of rigid body

Integrating the weighted average of positions over continuous mass distribution.

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Angular velocity

Rate of change of angular displacement with respect to time; all parts of a rotating body have the same angular velocity.

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Linear velocity

Velocity of a particle in a rigid body rotating about a fixed axis; calculated by multiplying the angular velocity by the distance from the axis of rotation (v = rω).

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Torque

Rotational equivalent of force; calculated by the cross product of the position vector and the force vector (τ = r × F).

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Angular momentum

Measure of a rotating object's rotational inertia; calculated by the cross product of the position vector and linear momentum (L = r × p).

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Translational Equilibrium

State where the net force on a body is zero.

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Rotational Equilibrium

State where the net torque on a body is zero.

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Moment of Inertia

Measure of a body's resistance to rotational acceleration; analogous to mass in linear motion.

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Principle of Moments

When the algebraic sum of moments of all forces acting on a body about a fixed point is zero, the body is in rotational equilibrium.

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Centre of Mass

The average position of all the mass in a system.

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Calculating Centre of Mass

Find the weighted average position of all particles in a system using their masses.

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Velocity of Centre of Mass

The time derivative of the centre of mass position.

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Newton's Second Law (CM)

The total force on a system equals the mass of the system times the acceleration of the center of mass.

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Conservation of Linear Momentum

If no external forces act on a system, its total momentum remains constant.

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Cross Product

A vector operation resulting in a vector perpendicular to both operands.

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Vector Product Magnitude

The magnitude is |a||b|sinθ, where a and b are the magnitudes of the vectors and θ is the angle between them.

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Vector Product Direction

Determined by the right-hand rule, pointing in the direction of perpendicular between the original two vectors.

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Center of Mass (Circular Ring)

For a circular ring, the center of mass is located at a distance of half the radius from a reference point. (x-a)/2 on x-axis)

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Center of Mass (Circular Cone)

The center of mass of a hollow circular cone is centred on its geometry.

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Total Angular Momentum

Sum of angular momentum of all particles in a system. This includes both individual angular momentum from the rotating motion and the effect of each particle's linear motion on the total Angular momentum.

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Angular Momentum of a Particle

Measured by the cross product of the particle's position vector (from a reference point) and its momentum vector.

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Pure Rolling

Rolling motion without slipping, where the velocity of the point of contact between the rolling object and the surface is zero.

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Pure Sliding

Movement where the object slides along the surface and doesn't roll.

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Rolling Acceleration

Acceleration of a rolling object down an incline, influenced by gravity and the object's moment of inertia.

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Minimum Friction Coefficient for Rolling

The lowest value of the friction coefficient needed to prevent slipping during rolling down an inclined plane.

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Rolling Motion Acceleration (No Slipping)

Acceleration of a rolling object when friction prevents slipping, influenced by gravity, moment of inertia.

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Slipping Motion Acceleration

Acceleration of a body sliding down an inclined surface with friction. The net acceleration accounts for both the component of gravity along the incline and the frictional force.

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