Problem 21 - Applied Physics

Question

The radius of a curve is \(27.5 \mathrm{~m}\). What is the centripetal force on a \(10, \overline{0} 00-\mathrm{kg}\) truck going around it at \(35.0 \mathrm{~km} / \mathrm{h}\) ?

Step-by-Step Solution

Answer

The centripetal force is approximately 34,356 N.

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

First, convert the speed of the truck from kilometers per hour to meters per second. We do this because standard units for calculations involving forces are meters per second (m/s) and kilograms (kg). Use the conversion factor:\[ 1 \, \text{km/h} = \frac{1000 \, \text{m}}{3600 \, \text{s}} = \frac{1}{3.6} \, \text{m/s} \]So, the speed \(35.0 \, \text{km/h}\) in meters per second is:\[ 35.0 \, \text{km/h} \times \frac{1}{3.6} = 9.72 \, \text{m/s} \]

Step 2: Write the Formula for Centripetal Force

The centripetal force \( F_c \) required to keep an object moving in a circle is given by:\[ F_c = \frac{m v^2}{r} \]where \( m \) is the mass of the truck, \( v \) is its velocity, and \( r \) is the radius of the curve.

Step 3: Substitute Known Values into the Formula

Now substitute the known values into the formula: - Mass \( m = 10,000 \, \text{kg} \)- Velocity \( v = 9.72 \, \text{m/s} \)- Radius \( r = 27.5 \, \text{m} \)\[ F_c = \frac{10,000 \times (9.72)^2}{27.5} \]

Step 4: Calculate the Centripetal Force

Calculate the expression:1. First compute \( v^2 \):\[ v^2 = (9.72)^2 = 94.4784 \, \text{m}^2/\text{s}^2 \]2. Multiply by the mass:\[ 10,000 \times 94.4784 = 944,784 \, \text{kg} \cdot \text{m}^2/\text{s}^2 \]3. Divide by the radius:\[ \frac{944,784}{27.5} = 34,355.78 \, \text{N} \]The centripetal force is approximately \( 34,355.78 \, \text{N} \).

Key Concepts

Understanding Circular Motion

In physics, circular motion refers to the movement of an object along the circumference of a circle. This motion can be either uniform or non-uniform, depending on whether the speed along the path is constant. When something moves in a perfect circle at a constant speed, we call it uniform circular motion.
Circular motion is fascinating because it involves a continuous change in direction. An object moving in a circle always has a force acting on it that pulls it inward, toward the center of the circle. This force is known as centripetal force, which is essential for maintaining the circular path. Without this force, the object would not be able to resist moving off in a straight line due to inertia.
Key points to remember about circular motion:

• The direction of velocity is always tangential to the circle.
• Centripetal force is directed toward the center of the circular path.
• Speed can be constant, but velocity is changing due to its changing direction.

Converting Velocity Units

In physics problems, it's crucial to work with the correct units. Converting units is a necessary step to ensure consistency and accuracy in calculations. For example, when dealing with velocity in the context of circular motion, you often need to convert km/h to m/s.
The conversion from kilometers per hour to meters per second is straightforward by using the conversion factor:\[ 1 \, \text{km/h} = \frac{1000 \, \text{m}}{3600 \, \text{s}} = \frac{1}{3.6} \, \text{m/s} \]So, when given a velocity in kilometers per hour, you multiply by \(\frac{1}{3.6}\) to convert it to meters per second. This is because using units of meters and seconds aligns with the SI (International System of Units), which is standard for most physics equations.
Always remember:

• Multiplying by \( \frac{1}{3.6} \) converts km/h to m/s.
• Converting ensures your velocity units match other SI units like meters and kilograms in equations.

Calculating the Centripetal Force

Calculating centripetal force involves a simple formula derived from Newton's Second Law of Motion:\[ F_c = \frac{m v^2}{r} \]This formula tells us that the required centripetal force is directly proportional to the mass of the object and the square of its velocity, and inversely proportional to the radius of the circular path.
Steps to calculate centripetal force:

• First, ensure you have the velocity in the correct units (usually m/s).
• Calculate the square of the velocity; remember, velocity needs to be squared for the equation.
• Multiply by the object's mass.
• Divide this product by the radius of the circle.

This force keeps your object moving in a circle without it flying away tangentially. Without adequate centripetal force, objects would move tangentially off the circle due to inertia.

Relating Mass and Radius in Physics

Mass and radius are integral components in the study of circular motion in physics. The mass of an object affects how much centripetal force is required to keep it moving along a circular path.
Similarly, the radius of the circle also plays a vital role. A larger radius means that, for the same velocity, you require less centripetal force. Conversely, a smaller radius demands more force for the same amount of velocity due to tighter turns.
Remember:

• The larger the mass, the greater the force needed to maintain circular motion.
• The larger the radius, the less force is needed for a given speed.
• Both mass and radius are key variables in determining the magnitude of the centripetal force.

Mass and radius hence are fundamental in the formula for calculating centripetal force and must be carefully considered when examining motion in circular paths.