Question

The M.I. of a body about the given axis is \(1.2 \mathrm{kgm}^{2}\) initially the body is at rest. In order to produce a rotational kinetic energy of \(1500 \mathrm{~J}\). an angular acceleration of \(25 \mathrm{rad} \mathrm{sec}^{2}\) must be applied about that axis for duration of \(\ldots\) \(\\{\mathrm{A}\\} 4 \mathrm{sec}\) \\{B \(\\} 2 \mathrm{sec}\) \(\\{C\\} 8 \mathrm{sec}\) \(\\{\mathrm{D}\\} 10 \mathrm{sec}\)

Short answer

The angular acceleration must be applied for a duration of \(2s\) to produce a rotational kinetic energy of \(1500J\). The correct answer is (B) \(2s\).

Step-by-step solution

Identify the given values

: Moment of inertia (I) = \(1.2 kgm^{2}\) Initial angular velocity (ω_initial) = \(0 rad/s\) (at rest) Target rotational kinetic energy (K.E.) = \(1500 J\) Angular acceleration (α) = \(25 rad/s^{2}\)

Use the rotational kinetic energy formula

: The formula for rotational kinetic energy is: K.E. = \(\frac{1}{2}Iω^{2}\) Where: K.E. is the rotational kinetic energy I is the moment of inertia ω is the final angular velocity We have been provided the target rotational kinetic energy (1500 J) and the moment of inertia (1.2 kgm²). Our goal is to find the final angular velocity (ω). Plug the known values into the equation: \(1500 = \frac{1}{2}(1.2)ω^{2}\)

Solve for final angular velocity

: Rearrange the equation and solve for ω: \(ω^{2} = \frac{3000}{1.2}\) Calculate ω: \(ω = \sqrt{\frac{3000}{1.2}} \approx 50 rad/s\)

Use the equation of motion for rotational motion

: The equation of motion for rotational motion is: \(ω = ω_{initial} + αt\) Where: ω is the final angular velocity ω_initial is the initial angular velocity α is the angular acceleration t is the time for which the angular acceleration is applied We know the initial angular velocity (0 rad/s), the final angular velocity calculated in step 3, and the angular acceleration (25 rad/s²). Now, we can solve for t: \(50 = 0 + (25)t\)

Solve for the duration

: Rearrange the equation and solve for t: \(t = \frac{50}{25}\) Calculate t: \(t = 2s\) So, the angular acceleration must be applied for a duration of \(2s\) to produce a rotational kinetic energy of \(1500J\). The correct answer is (B) \(2s\).

Key concepts

Moment of Inertia

Moment of inertia is a fundamental concept in the study of rotational motion. It describes how resistant an object is to changes in its rotational state, quite similar to the concept of mass in linear motion. For any object or system, the moment of inertia depends on the mass distribution relative to the axis about which it rotates. When calculating moment of inertia, you often express it in the form of \( I \), with units typically measured in \( kg \cdot m^2 \). The larger the moment of inertia, the more torque needed to change the rotational speed of the object.

• For simple geometric shapes, the moment of inertia can be calculated directly using standard formulas based on the shape and axis.
• For complex objects, it might be determined through integration or by summing up the contributions of individual parts.

In our exercise, the given moment of inertia is \(1.2 \ kg \cdot m^2\), indicating the level of resistance this body has to any rotational changes imposed on it.

Rotational Kinetic Energy

Rotational kinetic energy is the energy possessed by a rotating object due to its motion about an axis. It's given by the formula \( K.E. = \frac{1}{2}Iω^{2} \), where \( I \) is the moment of inertia and \( ω \) (omega) is the angular velocity.The energy can be seen as a measure of how much work a rotating object can do because of its motion. This concept parallels the kinetic energy in linear motion, but specifically accounts for rotation:

• A higher moment of inertia or higher angular velocity will result in a greater kinetic energy.
• Rotational kinetic energy can be converted to work or other forms of energy, and is crucial in analyzing systems like flywheels or turbines.

In the exercise, we aimed to achieve a rotational kinetic energy of \(1500 \ J\), which required manipulating the angular velocity given the moment of inertia.

Angular Acceleration

Angular acceleration describes how the angular velocity of an object changes with time. It is denoted as \( α \) and is measured in \( \, rad/s^2 \).When a torque is applied to a rotational system, it causes an angular acceleration that changes the angular velocity. The relationship between these quantities is found within Newton's second law for rotation.

• The formula to compute angular acceleration given the change in angular velocity and time is \( α = \frac{Δω}{Δt} \).
• Angular acceleration can be uniform or change over time, depending on the forces applied and the system setup.

In our case, the applied angular acceleration was \(25 \ rad/s^2\), which directly influenced the change in angular velocity necessary to achieve the desired kinetic energy.

Equations of Motion

Equations of motion for rotational dynamics are analogues to those used in linear motion but adapted to account for rotation. These equations use angular counterparts to linear velocity, acceleration, and displacement.A core equation is \( ω = ω_{initial} + αt \), relating the final angular velocity to the initial angular velocity through angular acceleration and time. This shows how time-dependent acceleration influences a rotating system.

• The concept of integration with respect to time bridges the rate of change (angular acceleration) and the result (angular velocity or displacement).
• Using these equations, we can predict the rotational behavior of objects under various conditions, much like with linear motion.

In the exercise, we applied the equation \( ω = ω_{initial} + αt \) to find the time duration necessary for the specified angular acceleration when transitioning from rest to a particular angular velocity. This ultimately led us to a duration of \(2 \ s\).