Initial Height h₁ m = 325 kg h₁ = 19.5 m 1 Car mass, with passengers = m No friction. 2 Bottom 3 Final Height h₂ h2=4.7 m A roller coaster car of mass m is rolling down the track, starting from rest, as shown in the figure. The initial height from the ground is h1 (Position-1), while the final height is h2 (Position-3). The system has no friction. What is the speed of the car when it reaches the bottom (Position-2) of the track.
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
![### Energy Conservation in a Roller Coaster Ride
#### Diagram Explanation:
A diagram of a roller coaster track is shown, divided into three key positions:
1. **Position 1 (Initial Height, \( h_1 \))**:
- The roller coaster starts at the topmost point of the track with an initial height \( h_1 = 19.5 \) meters from the ground.
2. **Position 2 (Bottom)**:
- The lowest point of the track, where we will calculate the speed of the roller coaster.
3. **Position 3 (Final Height, \( h_2 \))**:
- Another elevated point on the track where the final height \( h_2 \) is given as 4.7 meters.
- **Roller Coaster Car Information**:
- The mass of the roller coaster car, including passengers, is denoted as \( m = 325 \) kg.
- The track has no friction, implying that mechanical energy is conserved throughout the ride.
#### Problem Statement:
A roller coaster car of mass \( m \) is rolling down the track, starting from rest, as shown in the figure. The initial height from the ground is \( h_1 \) (Position-1), while the final height is \( h_2 \) (Position-3). The system has no friction.
What is the speed of the car when it reaches the bottom (Position-2) of the track?
#### Given:
- \( m = 325 \) kg
- \( h_1 = 19.5 \) m
- \( h_2 = 4.7 \) m
#### Solution Approach:
Using the conservation of mechanical energy principle:
1. **Potential Energy at Initial Point (Position 1)**:
- \( PE_1 = m \cdot g \cdot h_1 \)
2. **Kinetic Energy at Initial Point (Position 1)**:
- Since the car starts from rest, \( KE_1 = 0 \)
3. **Potential Energy at Bottom (Position 2)**:
- At the bottom, \( h = 0 \), so \( PE_2 = 0 \)
4. **Kinetic Energy at Bottom (Position 2)**:
- \( KE_2 = \frac{1}{2} m v^2 \), where \( v \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe77e1f08-afe7-4a66-b3a4-8fc9f2c917dc%2F45827bd0-e534-4938-9f83-f0f1f812ad66%2Fycdq0zs_processed.png&w=3840&q=75)
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