Question

Two objects suspended from identical springs are set into motion. The weight of one object is twice the weight of the other. How are the periods of the resulting motions related?

Short answer

Answer: The period of the resulting motion of the object with twice the weight is √2 times the period of the object with half the weight.

Step-by-step solution

Writing the period formulas for both objects.

For the object with mass m₁, the formula for its period T₁ is: T₁ = 2π √(m₁/k) For the object with mass m₂ (which is twice the mass of m₁), the formula for its period T₂ is: T₂ = 2π √(m₂/k)

Substituting m₂ = 2m₁ into the second formula.

Since m₂ = 2m₁, we can substitute this into the T₂ formula to find the relationship between T₁ and T₂: T₂ = 2π √((2m₁)/k)

Comparing the two formulas.

Now, let's compare the two T formulas to find the relationship between them. From step 1, we have: T₁ = 2π √(m₁/k) And from step 2, we have: T₂ = 2π √((2m₁)/k) Observing these two equations, we can see that their ratio is: T₂ / T₁ = √((2m₁)/k) / √(m₁/k) = √(2m₁/m₁) = √2 So, T₂ = T₁√2. This means that the period of the resulting motion of the object with twice the weight is √2 times the period of the object with half the weight.

Key concepts

Differential Equations

Differential equations are mathematical tools used to describe the relationship between functions and their derivatives, reflecting how an applied scenario changes over time. In the context of harmonic motion, such as that of objects attached to springs, we use second-order differential equations to model the restoring force proportional to their displacement.

This follows Hooke's Law, which states that the force \( F \) exerted by a spring is negative (since it is restoring) and proportional to the displacement \( x \) from the equilibrium position by the constant \( k \) (spring constant), or \( F = -kx \). When an object of mass \( m \) is attached to such a spring, we arrive at \( ma = -kx \) which simplifies to the classic second-order differential equation for simple harmonic motion: \( m\frac{d^2 x}{dt^2} = -kx \).

The solutions to these equations describe the position \( x \) of the mass at any time \( t \) and typically involve trigonometric functions such as sine and cosine, which outline the cyclical nature of the motion.

Spring Mass System

The spring mass system is an idealized model often used in physics to illustrate simple harmonic motion. This model comprises a mass \( m \) that is suspended from or attached to a spring with a known spring constant \( k \) and no other forces, like friction or air resistance, acting on it. The mass will oscilate back and forth once displaced from its equilibrium position.

Using Newton's second law \( F = ma \) and Hooke's Law \( F = -kx \), we can derive the equation of motion for a spring mass system, which is the second-order differential equation mentioned earlier. The motion's period \( T \) is given by the formula \( T = 2\pi\sqrt{\frac{m}{k}} \), asserting the period of oscilation solely depends on the mass and the spring constant. In our exercise, we considered two different masses and discerned how this relationship affected their periods. The system's inherent properties dictate the frequency of oscillation, making the spring mass system a fundamental example of harmonic motion.

Boundary Value Problems

Boundary value problems (BVPs) are types of differential equations where we are provided with certain conditions, called 'boundary conditions', at various points. In most physical scenarios, these conditions determine the unique solution to the differential equations since initial-value problems are not sufficient to define the motion's nature at all points.

In the case of the spring mass system, the boundary conditions could include initial displacement and velocity. While our exercise does not directly engage with boundary value problems, understanding them is crucial for dealing with more complex systems where initial conditions aren't the only concerns. For instance, in a system with multiple springs and masses or in various physical conditions such as varying temperatures, BVPs become indispensable for finding precise solutions.