https://file.aiquickdraw.com/mathbot/file/1727602600187u1l7cls4.jpg

To solve this problem, we need to determine which components contribute to the block's centripetal acceleration in the rotating inverted cone.

### Analyzing the diagram:
- The forces acting on the block are:
  - \( w \) is the gravitational force (weight).
  - \( n \) is the normal force from the cone's surface.
  - \( f \) is the frictional force preventing the block from sliding down.

The block undergoes centripetal acceleration due to the rotation of the cone, and this acceleration points toward the axis of rotation. For centripetal acceleration, we are concerned with the horizontal components of the forces, as they contribute to the inward radial force.

### Components of Forces:
1. **Normal Force (\(n\)):**
   The normal force \(n\) is directed perpendicular to the cone's surface. We can resolve it into two components:
   - One along the horizontal (radial) direction, which is \(n \sin \theta\).
   - One along the vertical direction, which is \(n \cos \theta\).

2. **Friction Force (\(f\)):**
   The frictional force \(f\) is along the surface of the cone and also has two components:
   - One along the horizontal (radial) direction, which is \(f \cos \theta\).
   - One along the vertical direction, which is \(f \sin \theta\).

Since centripetal acceleration is directed horizontally, the components of the forces that contribute to it will be the horizontal components:
- The horizontal component of the normal force is \( n \sin \theta \).
- The horizontal component of the friction force is \( f \cos \theta \).

### Answer:
Thus, the total force contributing to the centripetal acceleration is the difference of these two components:
\[ n \sin \theta - f \cos \theta. \]

This matches the option:
\[
\boxed{n \sin \theta - f \cos \theta}.
\]

---

Do you want further details or clarifications on any part of this solution? 

Here are some related questions you might want to explore:
1. How do we derive the expression for centripetal force in a rotating system?
2. What role does friction play in preventing the block from sliding?
3. How does the angle \( \theta \) affect the magnitude of the centripetal force?
4. Can the block slide down if the rotation speed is increased?
5. What happens if the surface of the cone is smooth (i.e., no friction)?

**Tip:** When analyzing forces in rotating systems, always separate the components of forces into radial (toward the center of rotation) and vertical (parallel to gravity) directions.

A rough inverted cone rotates at a constant speed about its vertical axis slowly enough so a block inside does not slide down. Given the block’s free-body diagram, which of the following components contribute to the block’s centripetal acceleration?

Learn how to solve a problem involving centripetal force acting on a block in a rotating inverted cone. This problem uses vector components, friction, and normal force, suitable for high school physics students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation