Question

Identical blocks oscillate on the end of a vertical spring one on Earth and one on the Moon. Where is the period of the oscillations greater? a) on Earth d) cannot be determined b) on the Moon from the information given c) same on both Farth and Moon

Short answer

Answer: c) The period of oscillations is the same on both Earth and the Moon.

Step-by-step solution

Understand the formula for the period of oscillations

The period of oscillations for a spring-block system is given by the formula T = 2 * pi * sqrt(m/k), where m is the mass of the block, k is the spring constant, and pi is a mathematical constant. This formula is derived from the equation for simple harmonic motion, which describes the motion of the block attached to the spring.

Compare the mass of the block on Earth and the Moon

Since the blocks are identical, their mass is the same on Earth and the Moon. Thus, the mass term in the formula for the period of oscillation does not affect the comparison between the periods on Earth and the Moon.

Compare the spring constant on Earth and the Moon

The spring constant k is a property of the spring itself and is not affected by the location or the presence of gravitational force. Since the identical blocks are attached to the same type of spring in both cases, the spring constant is the same on Earth and the Moon.

Compare the effect of gravity on Earth and the Moon

The period of oscillation formula does not explicitly include the gravitational force. Since the masses and spring constants are equal on Earth and the Moon, and gravity does not appear in the formula, there is no dependence on the gravitational force. Hence, the effect of gravity on Earth and the Moon does not change the period of oscillations.

Determine where the period of oscillations is greater

Since the masses and spring constants are equal on Earth and the Moon, and the gravitational force does not affect the period of oscillation, we can conclude that the period of oscillations is the same on Earth and the Moon. Therefore, the correct answer is c) same on both Earth and Moon.

Key concepts

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. This kind of motion is ubiquitous in the world of physics and is the foundation for understanding a variety of oscillating systems, from pendulums to the vibrations of atoms in a solid.

An object in simple harmonic motion has a special characteristic - its motion can be described by a sine or cosine function, which reflects the repetitive and smooth oscillations back and forth around a central point, called the equilibrium position. The formula for the period of these oscillations is \( T = 2\pi\sqrt{\frac{m}{k}} \) where \( T \) is the period, \( m \) is the mass of the oscillating body, and \( k \) is the spring constant.

SHM is crucial in understanding physics because it represents the purest form of oscillatory movement. By studying SHM, we can more easily comprehend other more complex motions which are often combinations or variations of simple harmonic motion.

Spring-Block System

The spring-block system is a classic example of simple harmonic motion and serves as an ideal model for studying the period of oscillations. The setup usually consists of a block of mass \( m \) attached to a spring with a spring constant \( k \) which governs the stiffness of the spring. When the block is pulled and released, it oscillates back and forth along the axis of the spring.

The period \( T \) of these oscillations is independent of the amplitude, meaning that regardless of how far the block is initially pulled, the time it takes to complete one full cycle of motion remains constant, provided that the system is undamped and that the amplitude is not too large. Interestingly, because the period relies only on the mass and the spring constant, variations in other factors, like gravitational acceleration, do not affect the period in the ideal spring-block system.

One important note when considering a spring-block system is the assumption of an ideal spring that obeys Hooke's law, which posits that the force exerted by the spring is proportional to the displacement. This linear relationship holds true up to a certain point beyond which the spring does not behave elastically - a limitation often excluded in introductory physics discussions.

Gravitational Force

Gravitational force is a universal force of attraction that exists between any two bodies with mass. It plays a significant role in the orbits of planets, the tides in oceans, and even the flight of a ball thrown into the air. The force of gravity on an object is proportional to its mass and the mass of the body it is near, and inversely proportional to the square of the distance between the centers of masses of these two bodies, as described by Newton's law of universal gravitation.

Despite its universal presence, gravity does not directly enter the formula for the period of simple harmonic motion in a spring-block system, as seen in the example problem. A common misconception is that gravity alters the period of an object's oscillation because it affects its weight, or the force due to gravity acting on it. Weight and mass are proportionally related; as such, the gravitational force does impact the weight of the block in a spring-block system, but because the period of oscillations formula depends on mass, not weight, gravity does not influence the period. This constancy reaffirms that the simple harmonic motion seen in an ideal spring-block system is consistent across different gravitational conditions; therefore, an object will oscillate with the same period on the moon as it will on earth, assuming identical systems.