GTU Physics Paper

Explanation: The question asks how a 20% increase in the angular speed of a rotating body affects its rotational kinetic energy. Rotational kinetic energy depends on the square of angular velocity, meaning any change in speed affects energy quadratically. When angular velocity increases, kinetic energy rises by a factor proportional to the square of the new angular speed divided by the old one. This relationship can be calculated step by step: first, express the new angular velocity as 1.2 times the original, then square this factor to find the increase in kinetic energy. For example, if a wheel rotates faster, the energy required grows faster than the speed itself due to the squared dependence. In essence, even a moderate increase in angular speed leads to a noticeably larger increase in rotational energy. This highlights the quadratic relationship between rotational speed and energy in Rotational Dynamics. The key idea is that doubling speed would quadruple energy, and small increments still produce larger percentage changes in energy.

Explanation: This question examines the relationship between changes in angular velocity, kinetic energy, and angular momentum. Angular momentum depends linearly on angular velocity, while kinetic energy depends on the square of angular velocity. Given a known increase in kinetic energy when angular speed doubles, the corresponding change in angular momentum can be analyzed step by step. First, determine how angular velocity changes relative to initial velocity. Then, use the linear proportionality of angular momentum to angular speed to compute the expected change. This approach highlights the difference between linear and quadratic dependencies in rotational motion. For instance, if a spinning disk gains energy, the momentum increases proportionally but not as sharply as energy due to the squared relationship. Understanding this distinction is crucial in solving rotational mechanics problems involving simultaneous changes in speed, energy, and momentum. In short, kinetic energy grows faster than angular momentum when speed increases.

Explanation: This question asks to determine the radius of gyration of a flywheel when work is done to change its angular speed. work done on a rotating body changes its rotational kinetic energy, which is given by ½ Iω², where I = m·k² is the moment of inertia expressed via radius of gyration k. Step by step, first convert angular speeds from rpm to rad/s. Then, relate the work done to the change in rotational kinetic energy: ΔK = W = ½ I (ω₂² – ω₁²). Substitute I = m·k² and solve for k. This method shows how energy considerations determine structural parameters of rotating bodies. For instance, a spinning wheel absorbs work into kinetic energy, and its effective radius determines how Mass is distributed relative to the axis. Overall, radius of gyration quantifies how Mass affects rotational energy for a given speed change.

Explanation: The problem examines how a percentage change in angular velocity affects rotational kinetic energy. Rotational kinetic energy depends on the square of angular velocity, K = ½ Iω². Therefore, a small change in ω results in a larger proportional change in K. Step by step, represent the new angular speed as 1.2 times the original. Then, compute the square of this factor to find the corresponding energy increase: (1.2)² = 1.44, meaning kinetic energy increases by 44%. For example, if a spinning disk rotates faster, the energy rises faster than the speed itself due to the squared relationship. This highlights that energy scales quadratically with angular speed, not linearly. In short, even modest increases in angular speed require significantly more energy.

Explanation: This question explores the link between angular velocity and rotational kinetic energy. Rotational kinetic energy is proportional to the square of angular speed, K = ½ Iω². Thus, a 10% increase in ω (factor of 1.1) will increase K by (1.1)² – 1 = 0.21, or 21%. Step by step, first express the new angular velocity as 1.1 times the original. Then square it to determine the new energy factor, subtract the original to get the percentage increase. An analogy is speeding up a spinning fan: a small increase in rotation results in a noticeably larger increase in stored rotational energy due to the squared dependence. In summary, kinetic energy grows faster than angular speed, illustrating the nonlinear relationship in Rotational Dynamics.