Problem 18 - Applied Physics

Question

A truck with mass 215 slugs rounds a curve of radius \(53.0 \mathrm{ft}\) with a speed of \(62.5 \mathrm{ft} / \mathrm{s} .\) (a) What centripetal force is exerted on the truck while rounding the curve? (b) How does the centripetal force change when the velocity is doubled? (c) What is the new force?

Step-by-Step Solution

Answer

(a) 15837.64 pounds, (b) Centripetal force quadruples, (c) 63350.56 pounds.

Step 1: Understand the formula for centripetal force

The formula for centripetal force \( F_c \) is given by \[ F_c = \frac{mv^2}{r} \]where \( m \) is the mass of the truck, \( v \) is the velocity, and \( r \) is the radius of the curve.

Step 2: Calculate the centripetal force

Substitute the given values into the formula: - Mass \( m = 215 \) slugs,- Velocity \( v = 62.5 \) ft/s,- Radius \( r = 53.0 \) ft.Calculating, we get\[ F_c = \frac{215 \times (62.5)^2}{53} \] \[ F_c = \frac{215 \times 3906.25}{53} \] \[ F_c \approx \frac{839343.75}{53} \] \[ F_c \approx 15837.64 \] Thus, the centripetal force exerted on the truck is approximately 15837.64 pounds.

Step 3: Assess the effect of doubling the velocity on the centripetal force

When the velocity \( v \) is doubled, it becomes \( 2v \). The new centripetal force \( F_c' \) is then given by:\[ F_c' = \frac{m(2v)^2}{r} = \frac{m \times 4v^2}{r} = 4 \times \frac{mv^2}{r} \]Thus, doubling the velocity quadruples the centripetal force.

Step 4: Calculate the new centripetal force

Since the new centripetal force \( F_c' \) is four times the original, we calculate it as:\[ F_c' = 4 \times 15837.64 \approx 63350.56 \]Therefore, the new force is approximately 63350.56 pounds.

Key Concepts

Physics Education

Physics education serves as the foundation for understanding the intricate details of natural phenomena, including concepts like centripetal force. It's all about making abstract ideas tangible and understandable.
The core of physics education lies in breaking down complex problems into digestible segments. For instance, understanding how different variables such as mass, velocity, and radius influence forces acting on bodies in motion.
When we explore the relation between these variables in physics exercises, we apply concepts from the textbook to real-world scenarios. This not only enhances comprehension but also develops problem-solving skills, critical thinking, and the ability to apply theoretical knowledge to practical situations.

Circular Motion

Circular motion is a fundamental concept in physics where an object moves along the circumference of a circle. This type of motion is prevalent in many aspects of day-to-day life, from cars rounding curves to planets orbiting the sun.
A crucial part of circular motion is understanding **centripetal force**. This is the inward force that keeps an object moving in a circular path. Without it, objects would simply move in a straight line due to inertia.

• Key components include the object's mass, its velocity, and the radius of the circle along which it travels.
• These components are interlinked, as changes in one can affect the others, influencing how much centripetal force is needed.
• For instance, an increase in velocity results in a substantial increase in the required centripetal force to maintain the circular path.

Understanding these relationships helps predict how objects will behave in circular paths, a foundation in physics study and practical application.

Velocity and Force Relationship

In the context of circular motion, the relationship between velocity and force is intricate yet fascinating.
Velocity affects the centripetal force significantly. Since centripetal force is calculated as \(F_c = \frac{mv^2}{r}\), the velocity component \(v^2\) indicates that any change in velocity causes a squared change in force.

• Doubling the velocity results in quadrupling the centripetal force. This is because the force depends on the square of the velocity.
• This sensitivity to velocity changes makes understanding this relationship critical, especially in fields like automotive safety and mechanical engineering.
• For students, grasping this concept can simplify complex physics problems by focusing on critical variables affecting motion and force.

Ultimately, this relationship illustrates how interconnected forces and motion are, forming the basis for much of physics application and theory.