Problem 17 - Applied Physics

Question

What is the centripetal force on a \(75 \overline{0}-\mathrm{kg}\) vehicle rounding a curve of radius \(40.0 \mathrm{~m}\) at \(30.0 \mathrm{~km} / \mathrm{h} ?\)

Step-by-Step Solution

Answer

The centripetal force is approximately 1301 N.

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

First, convert the velocity of the vehicle from kilometers per hour to meters per second. The conversion factor is 1 km/h = 0.2778 m/s. Therefore, \[v = 30.0 \, \text{km/h} \times 0.2778 \, \text{m/s per km/h} \approx 8.33 \, \text{m/s}\]

Step 2: Use the formula for centripetal force

Centripetal force \( F_c \) is given by the formula:\[F_c = \frac{mv^2}{r}\]where \( m \) is the mass of the vehicle, \( v \) is the velocity in m/s, and \( r \) is the radius of the curve. Given \( m = 750 \, \text{kg} \), \( v = 8.33 \, \text{m/s} \), and \( r = 40.0 \, \text{m} \), substitute these values into the formula:\[F_c = \frac{750 \, \text{kg} \times (8.33 \, \text{m/s})^2}{40.0 \, \text{m}}\]

Step 3: Calculate the value of centripetal force

Calculate the value using the values inserted into the formula:\[F_c = \frac{750 \, \text{kg} \times 69.39 \, \text{m}^2/\text{s}^2}{40.0 \, \text{m}} = \frac{52042.5 \, \text{kg} \cdot \text{m/s}^2}{40.0 \, \text{m}} = 1301.06 \, \text{N}\]

Step 4: Finalize and round the result

Round the calculated centripetal force to an appropriate number of significant figures, usually the same as the least precise value given in the problem data:\[F_c \approx 1301 \, \text{N}\]Therefore, the centripetal force on the vehicle is approximately 1301 Newtons.

Key Concepts

Conversion of Velocity Units

When dealing with questions of motion, like calculating centripetal force, having the correct units for velocity is essential. Often, velocities are given in kilometers per hour (km/h), but for physics calculations, such as those involving forces, it's often more useful to work in meters per second (m/s). This ensures consistency with other units like meters for distance and seconds for time.
To convert from km/h to m/s, we use the fact that 1 km/h is equivalent to approximately 0.2778 m/s. Thus, to convert 30.0 km/h, you multiply by 0.2778:
- 30.0 km/h × 0.2778 = 8.33 m/s
This conversion simplifies further calculations by aligning all units properly. Always ensure you're working in consistent units to avoid errors in physics problem-solving.

Formula for Centripetal Force

Centripetal force is what keeps an object moving in a circular path, acting towards the center of the circle. To calculate this force, we use the formula:
\[F_c = \frac{mv^2}{r}\]
Here, \( F_c \) represents the centripetal force, \( m \) is the mass of the object, \( v \) is the velocity, and \( r \) is the radius of the circle. It's important to remember that velocity must be squared, which sometimes catches students off-guard.
Imagine pushing a car around a curve; to keep it on the path, it must constantly be redirected. That's essentially what centripetal force does!

Physics Problem-Solving

Solving physics problems often involves several steps to get a precise answer. Start by carefully understanding the problem statement, identifying known quantities (like mass, speed, radius), and what needs to be calculated (in this case, centripetal force).
After unit conversions (like converting velocity from km/h to m/s), substitute the known values into the relevant formulas. It helps to break the problem down into smaller tasks, simplifying complex equations incrementally. Double-check each step as you go, ensuring everything is correctly applied.
Lastly, establish a systematic approach. This enhances accuracy and builds confidence, helping you tackle more complex problems effectively.

Significant Figures in Calculations

Significant figures are crucial in physics to express a numerical answer to its appropriate precision. They reflect the precision of the measurements used in a calculation. With each step of solving, always consider how many significant figures the initial values carry.
For example, in the centripetal force calculation, the radius and velocity were given to three significant figures. Thus, the final answer should also reflect the same precision level—hence 1301 N rather than more exact decimals like 1301.06 N.
Using proper significant figures isn't just a formality; it communicates the certainty of your results and ensures that your answers are scientifically valid and meaningful.