Problem 9 - An Introduction to Mechanics

Question

A block of mass \(M\) on a horizontal frictionless table is connected to a spring (spring constant \(k\) ). The block is set in motion so that it oscillates about its equilibrium point with a certain amplitude \(A_{0}\). The period of motion is \(T_{0}=2 \pi \sqrt{M / k}\). (a) A lump of sticky putty of mass \(m\) is dropped onto the block. The putty sticks without bouncing. The putty hits \(M\) at the instant when the velocity of \(M\) is zero. Find (1) The new period. (2) The new amplitude. (3) The change in the mechanical energy of the system. (b) Repeat part ( \(a\) ), but this time assume that the sticky putty hits \(M\) at the instant when \(M\) has its maximum velocity.

Step-by-Step Solution

Answer

(1) New period is longer: \( T_{1} = 2\pi \sqrt{\frac{M+m}{k}} \). (2) Amplitude is unchanged when velocity is zero, reduced when at max velocity. (3) No energy change at zero velocity, reduced at max velocity.

Step 1: Identifying conditions for new period

The block has an initial period of motion given by the formula \( T_{0} = 2\pi \sqrt{\frac{M}{k}} \). When additional mass \( m \) is added to mass \( M \) while the velocity is zero, only the mass affects the period. Calculate the new period by substituting total mass \( M+m \) into the period formula.

Step 2: Calculate New Period when velocity is zero

The new period \( T_{1} \) is given by \( T_{1} = 2\pi \sqrt{\frac{M+m}{k}} \), which is longer than \( T_{0} \) because the mass is increasing.

Step 3: Determining resulting amplitude when velocity is zero

The amplitude remains unchanged since the putty sticks to the block at the equilibrium position where the velocity is zero but maximum potential energy is stored.

Step 4: Energy change when putty hits at zero velocity

The total mechanical energy of the system does not change because the putty is added when the potential energy and not kinetic energy are at maximum, i.e., \( E = \frac{1}{2}kA_{0}^{2} \).

Step 5: Identifying conditions for maximum velocity

At maximum velocity, the block has maximum kinetic energy, and the potential energy at this moment is zero. After the collision, momentum is conserved, allowing us to determine the new velocity and energy of the system.

Step 6: New Period when putty hits at maximum velocity

Since period only depends on mass and spring constant, the new period remains \( T_{1} = 2\pi \sqrt{\frac{M+m}{k}} \) regardless of initial velocity conditions.

Step 7: Calculate new velocity and amplitude

At maximum velocity \( V_{max} = A_{0}\sqrt{\frac{k}{M}} \), conservation of momentum gives \( (M+m)v' = Mv_{max} \). Solve for \( v' \) to find new amplitude \( A' = v'\sqrt{\frac{(M+m)}{k}} \), which is reduced from \( A_{0} \).

Step 8: Energy change when putty hits at maximum velocity

Kinetic energy changes because apparent velocity decreases: \( \frac{1}{2}(M+m)v'^2 \) is less than \( \frac{1}{2}MV_{max}^2 \), showing a decrease in total mechanical energy after the collision due to conversion to heat.

Key Concepts

Oscillations

Oscillations are repetitive variations, typically in a physical system, where an object moves to and fro about an equilibrium position. This movement involves alternating kinetic and potential energies. In the context of a spring-mass system, when a block is attached to a spring and displaced from its resting position, it will start moving back and forth around the equilibrium point. Due to the system's nature, two important energies are in play:

• Kinetic energy, which is highest when the block moves fastest at the equilibrium position.
• Potential energy, stored in the spring when the block is at its maximum displacement.

These energies transform back into each other, creating a continuous oscillation. The study of such oscillations is crucial in understanding systems in harmonic motion, where predictable patterns repeat over time. Key parameters include frequency and period, which describe how often oscillations occur.

Spring-Mass System

A spring-mass system is a classic example of simple harmonic motion, where a spring's restoring force keeps a mass in oscillation. The system is defined by several characteristics:

• Mass (M) attached to the spring
• Spring constant (K) that indicates the stiffness of the spring
• Natural period (T) that determines the oscillation duration

The period of oscillation in a frictionless spring-mass system is determined by the formula \(T = 2\pi \sqrt{\frac{M}{K}}\).This means that the time it takes to complete one full back-and-forth motion depends only on M and K.Adding mass to the system increases the period, making the oscillations slower. This is because more mass requires greater force and time to achieve similar displacements. Even when an external force, like a lump of putty, is added, the spring-mass system continues oscillating, albeit with modified parameters such as period or amplitude. The simplicity and predictability of the spring-mass system make it widely useful in studying not only physical systems but also concepts of damping and resonance.

Mechanical Energy Conservation

In a spring-mass system, mechanical energy is conserved unless external forces, such as friction or non-conservative forces, intervene. Mechanical energy comprises:

• Potential energy, stored in the spring when it's stretched or compressed.
• Kinetic energy, associated with the moving mass.

When discussing conservation, it's essential to consider the law of energy conservation. As the block oscillates, energy shifts from potential to kinetic and back. This conservation of energy ensures that in ideal conditions (like in a frictionless environment), the system keeps oscillating indefinitely with constant energy E, given by\(E = \frac{1}{2}K A^{2}\).However, external interventions, such as when a putty is added at maximum velocity, disrupt this balance. The system loses some stored energy due to inelastic collisions and transformations into other forms of energy, such as heat. This results in a reduced amplitude and overall mechanical energy in the system, emphasizing that real systems often deal with energy dissipation.