Problem 16 - Applied Physics

Question

What is the centripetal force on a \(15 \overline{0} 0-\mathrm{kg}\) vehicle driven around a curve of radius \(35.0 \mathrm{~m}\) at \(60.0 \mathrm{~km} / \mathrm{h}\) ?

Step-by-Step Solution

Answer

The centripetal force is 11918.1 N.

Step 1: Convert Speed from km/h to m/s

To convert the speed from km/h to m/s, use the conversion factor: \(1 \, \text{km/h} = \frac{1}{3.6} \, \text{m/s}\). Therefore, the speed \(v\) is given by: \[v = 60.0 \, \text{km/h} \times \frac{1}{3.6} = 16.67 \, \text{m/s}\]

Step 2: Identify Known Quantities

We have the following known quantities: mass \(m = 1500 \, \text{kg}\), radius of the curve \(r = 35.0 \, \text{m}\), and speed \(v = 16.67 \, \text{m/s}\).

Step 3: Use the Formula for Centripetal Force

The centripetal force \(F_c\) is calculated using the formula: \[ F_c = \frac{m v^2}{r} \] where \(m\) is mass, \(v\) is velocity, and \(r\) is the radius of the curve.

Step 4: Calculate the Centripetal Force

Substitute the known values into the centripetal force formula: \[ F_c = \frac{1500 \, \text{kg} \times (16.67 \, \text{m/s})^2}{35.0 \, \text{m}} \] \[ F_c = \frac{1500 \, \text{kg} \times 278.0889 \, \text{m}^2/ ext{s}^2}{35.0 \, \text{m}} \] \[ F_c = \frac{417133.35 \, \text{kg} ext{ m/s}^2}{35} = 11918.1 \, \text{N} \]

Step 5: Final Answer

The centripetal force exerted on the vehicle is approximately: \[ F_c = 11918.1 \, \text{N} \].

Key Concepts

Circular Motion

When we talk about circular motion, we refer to the movement of an object along the circumference of a circle. It's essential to understand a few key terms to grasp the concept fully.

• Radius: This is the distance from the center of the circle to any point on its perimeter. In our problem, it's given as 35.0 meters.
• Circumference: This is the entire path that an object follows in its circular motion; for a complete circle, it's calculated as \(2\pi r\).
• Velocity: In circular motion, the speed is constant, but the direction is continuously changing. Therefore, the object is always accelerating.

This changing direction gives rise to a centripetal ("center-seeking") force which keeps the object moving in a circle. Without this force, the object would move off in a straight line. Remember, centripetal force itself is not a new type of force; it's usually the result of gravitational, tension, frictional, or any other type of force acting sideways to the direction of motion.

Physics Problem Solving

Solving physics problems involves understanding the underlying principles and applying them to find the solution. It starts with grasping the problem statement and identifying what is given and what needs to be determined. In this centripetal force problem, the goal is to calculate the force that keeps a vehicle moving in a circular path.

• Analyzing the Information: Determine what is already provided, such as the mass of the vehicle, radius of the curve, and initial speed.
• Using Formulas: Apply relevant physics equations, such as the formula for centripetal force, \( F_c = \frac{m v^2}{r} \). Each symbol in the formula represents a physical quantity, essential to solving the problem.
• Checking Units: Ensure that all values are in the correct units before inserting them into the equation. This often involves converting units.

Approach the problem methodically for an accurate and logical solution. Understanding the relationships between different variables in the problem helps in applying the correct formula to achieve the desired results.

Force Calculation

Calculating forces involves determining how much push or pull is exerted on an object to keep it in motion or at rest. The centripetal force calculation here uses the formula \( F_c = \frac{m v^2}{r} \).

• Mass \((m)\): Tells us how much matter is in the vehicle. Here it's 1500 kg.
• Velocity \((v)\): Represents the speed of the vehicle in a specified direction. After conversion, this speed is 16.67 m/s.
• Radius \((r)\): This is 35.0 meters in this instance, showing how tightly the curve is the vehicle is traveling on.

By plugging in these values into the formula, we can calculate centripetal force:
\( F_c = \frac{1500 \, \text{kg} \times (16.67 \, \text{m/s})^2}{35.0 \, \text{m}} \approx 11918.1 \, \text{N} \).
This represents the inward force necessary to keep the vehicle moving in its circular path.

SI Units Conversion

Understanding how to convert units is vital in physics, making sure measurements are consistent and can be appropriately used in equations. In our example, we need to convert speed from kilometers per hour to meters per second.

• Kilometers to Meters: Since 1 kilometer equals 1000 meters, multiply the kilometer value by 1000.
• Hours to Seconds: There are 3600 seconds in an hour, so divide the speed by 3.6 to convert from km/h to m/s.

Our exercise uses the conversion factor: \(1 \, \text{km/h} = \frac{1}{3.6} \, \text{m/s}\). Therefore, for a speed of 60.0 km/h, it becomes \(60.0 \, \text{km/h} \times \frac{1}{3.6} = 16.67 \, \text{m/s}\).
Accurate unit conversion is crucial for precise force calculations, ensuring that the numbers used in equations correspond to the correct physical quantities.