Question

A 0.800-kg ornament is hanging by a 1.50-m wire when the ornament is suddenly hit by a 0.200-kg missile traveling horizontally at 12.0 m/s. The missile embeds itself in the ornament during the collision. What is the tension in the wire immediately after the collision?

Short answer

The tension in the wire immediately after the collision is 13.64 N.

Step-by-step solution

Understand the Problem

We need to find the tension in the wire immediately after the collision between a missile and an ornament. The missile embeds itself into the ornament, making this an inelastic collision.

Apply Conservation of Momentum

In inelastic collisions, linear momentum is conserved. The initial momentum of the system is the momentum of the missile, since the ornament is initially at rest. Use the formula: \[ p = m_{missile} \cdot v_{missile} = 0.200 \, \text{kg} \cdot 12.0 \, \text{m/s} \]Which gives the total initial momentum: \[ p_{initial} = 2.4 \, \text{kg} \cdot \text{m/s} \]

Find the Velocity After Collision

Since the total mass after the collision is the sum of both masses, use the conservation of momentum:\[ p_{final} = (m_{missile} + m_{ornament}) \cdot v_{final} \]Set \( p_{initial} = p_{final} \) and solve for \( v_{final} \):\[ 2.4 \, \text{kg} \cdot \text{m/s} = (0.200 + 0.800) \, \text{kg} \cdot v_{final} \]\[ v_{final} = \frac{2.4}{1.0} = 2.4 \, \text{m/s} \]

Identify Tension in the Wire

Immediately after the collision, the system's velocity causes centripetal force, which contributes to tension in the wire. The system's mass is 1.0 kg, and the wire's radius is 1.5 m.

Calculate Tension Using Centripetal Force Formula

The tension in the wire is the sum of the centripetal force and the gravitational force acting on the ornament. Calculate the centripetal force: \[ F_c = \frac{m_{total} \cdot v_{final}^2}{r} = \frac{1.0 \, \text{kg} \cdot (2.4 \, \text{m/s})^2}{1.5 \, \text{m}} = 3.84 \, \text{N} \]Total tension T is \[ T = m_{total} \cdot g + F_c = (1.0 \, \text{kg} \cdot 9.8 \, \text{m/s}^2) + 3.84 \, \text{N} = 13.64 \, \text{N} \]

Conclude the Solution

The immediate tension in the wire after the collision is thus 13.64 N, which takes into account both the gravitational force and the induced centripetal force after the strike.

Key concepts

Conservation of Momentum

In physics, particularly in the study of collisions, the conservation of momentum plays a critical role. Momentum is a vector quantity defined as the product of an object's mass and velocity. In an isolated system, the total momentum remains constant if no external forces are acting.
After a collision, inelastic or elastic, the total initial momentum equals the total final momentum of the system. This exercise highlights an inelastic collision, where the missile embeds into the ornament.
Here's how you can calculate it:

• Determine the momentum of the moving object. Initially, the ornament is at rest, so only the missile has momentum.
• Use the equation for momentum: \( p = m \cdot v \) where \( m \) is mass and \( v \) is velocity.
• Calculate the initial momentum of the missile as it is the product of its mass (0.200 kg) and velocity (12 m/s), which gives 2.4 \( \text{kg} \cdot \text{m/s} \).
• After the collision, the ornament and missile combine to form one object, making the total mass 1.000 kg. Solve for the final velocity using the conserved momentum equation.

This principle allows us to find the speed of the combined mass immediately after collision, an essential step before considering other forces like tension.

Tension in a Wire

Tension in a wire arises from forces acting along the wire and is essential when analyzing the dynamics of coupled systems, like the ornament and missile scenario in our problem. After the collision, the system moves due to the momentum transferred.
The wire provides a centripetal force to keep the ornament in circular motion immediately after the collision. Tension in the wire has a dual responsibility here: it must counteract the weight of the ornament and provide the necessary centripetal force.
To calculate it, consider the contributions:

• The gravitational force, \( F_g = m_{total} \cdot g \), where \( g = 9.8 \, \text{m/s}^2 \).
• The centripetal force, \( F_c = \frac{m_{total} \cdot v_{final}^2}{r} \), helps keep the ornament moving along the circular path.

Summing these forces results in the total tension in the wire. Understanding tension in these contexts is key to mastering dynamics of motion in physics.

Centripetal Force

Centripetal force is the inward force required to keep an object moving in a circular path. In the case of the ornament and missile, after the collision, the system must move in a circular arc due to the tension in the wire.
This force acts perpendicular to the object's instantaneous velocity and towards the center of the path's curvature. It's essential to understand:

• Centripetal force is a result of other forces (e.g., tension, gravity), not a fundamental force itself.
• It depends on the object's mass, the square of its velocity, and the radius of its circular path.
• In formula form: \( F_c = \frac{m \cdot v^2}{r} \).

By computing centripetal force, we comprehend how much of the tension in the wire is needed just to maintain the object's circular motion. It's crucial for solving problems that involve rotation, orbit, or any kind of curvature in the path of motion.