Problem 98 - Physics for Scientists and Engineers with Modern Physics

Question

The sides of a cone make an angle \(\phi\) with the vertical. A small mass \(m\) is placed on the inside of the cone and the cone, with its point down, is revolved at a frequency \(f\) (revolutions per second) about its symmetry axis. If the coefficient of static friction is \(\mu_{\mathrm{s}}\), at what positions on the cone can the mass be placed without sliding on the cone? (Give the maximum and minimum distances, \(r\), from the axis).

Step-by-Step Solution

Answer

Find \( r \) values by solving \( \mu_s (m g \cos \phi + m r \omega^2 \cos \phi) = m r \omega^2 \sin \phi - mg \sin \phi \) for equilibrium.

Step 1: Analyze the Forces

Consider the forces acting on the mass: the gravitational force \( mg \), the centrifugal force \( m r \omega^2 \) (with \( \omega = 2 \pi f \)), and the normal force \( N \). The gravitational force can be resolved into two components: one perpendicular to the surface of the cone and one parallel to it. The centrifugal force is perpendicular to the axis of rotation.

Step 2: Find the Normal Force

The normal force \( N \) acts perpendicular to the cone's surface. We need to resolve the forces in the direction parallel to the surface and equate them to the frictional force. The component of the weight along the cone's surface is \( mg \sin \phi \) and needs to be balanced by the frictional force, which is \( \mu_s N \).

Step 3: Set Up Equilibrium Equations

In the direction along the surface of the cone, the frictional force \( f \) must balance the centrifugal force component \( m r \omega^2 \sin \phi \) and the component of gravitational force \( mg \sin \phi \). Equations can be set up as follows: \( \mu_s N = m r \omega^2 \sin \phi - mg \sin \phi \).

Step 4: Express Normal Force

The normal force \( N \) has components \( N = m g \cos \phi + m r \omega^2 \cos \phi \) when analyzing perpendicular to the surface of the cone.

Step 5: Solve for Radius \( r \)

Substitute for \( N \) from the perpendicular equilibrium into the friction equation: \( \mu_s (m g \cos \phi + m r \omega^2 \cos \phi) = m r \omega^2 \sin \phi - mg \sin \phi \). Simplifying this equation gives us the criteria for \( r \). Solve for \( r \) to find the maximum and minimum values allowed by the equilibrium condition.

Step 6: Interpret the Solution

The expression derived for \( r \) represents the range of distances from the axis of rotation where the block will not slide. Identify \( r_{\text{min}} \) and \( r_{\text{max}} \) by solving the quadratic condition given by setting the derivative with respect to \( r \) to zero and testing bounds of expression.

Key Concepts

Centrifugal Force

When dealing with rotational motion, centrifugal force is a crucial concept to understand. It often feels like a force that pushes objects outward as they rotate around a central point. However, it isn't an actual force in the traditional sense, but more of an apparent force due to inertia, as the object strives to move in a straight line due to its velocity.

Consider this scenario within a rotating cone. When a small mass is positioned on the inside surface of the cone, it experiences a centrifugal force given by the formula: \[ F_c = m r \omega^2 \]where \( m \) is the object's mass, \( r \) is the distance from the rotation axis, and \( \omega = 2 \pi f \) is the angular velocity, with \( f \) being the frequency of rotation. This force acts radially outward, perpendicular to the axis of rotation, attempting to push the mass away from the axis.

In essence, the role of centrifugal force in this scenario is to counterbalance the gravitational force that tends to pull the mass down the slope of the cone. Understanding this apparent force helps in setting up the equilibrium conditions necessary for calculating possible positions on the cone where the mass remains in static friction.

Equilibrium of Forces

In the setup of the rotating cone with a small mass inside, the concept of equilibrium of forces is fundamental. Equilibrium occurs when all the forces acting on the mass are perfectly balanced, ensuring the mass stays put on the surface of the cone without sliding.

Several forces play a role here and need to be carefully considered:

• The gravitational force \( mg \), which can be split into components: one tangential along the cone and another perpendicular to the cone's surface.
• The centrifugal force, \( m r \omega^2 \), acting outwardly from the cone's rotational axis.
• The normal force \( N \), which pushes directly against the mass perpendicular to the slanted surface of the cone.
• The frictional force that acts up the slope to stop sliding, described by \( f = \mu_s N \).

To solve for equilibrium, the forces in the tangential direction (down the slope) must balance. Therefore, the frictional force should equal the difference between the tangential component of the centrifugal force and the gravitational component in that direction. With this setup, we can establish the necessary equations to solve for the permissible positions \( r \) on the cone.

Cone and Mass Dynamics

The interaction between a rotating cone and a mass resting on its inside surface involves interesting dynamics. The motion and stability of the mass are influenced by the forces acting on it, combined with the geometry of the cone.

The cone makes an angle \( \phi \) with the vertical, and this angle plays a crucial role in determining the balance of forces. Understanding the dynamics involves analyzing how the cone's surface guides the forces acting on the mass.

The mass’s potential positions are primarily governed by static friction, which opposes sliding along the cone’s surface. When the cone rotates, the centrifugal force tends to make the mass slide up, while gravity tries to pull it down. Static friction helps keep the mass stable as long as these forces are balanced.

For different positions \( r \), the balance between centrifugal force and static friction defines the zone where the mass won't slide. By calculating the maximum and minimum \( r \) that satisfy these balance conditions, we determine the stable region where the mass can be placed on the cone without slipping. This understanding is essential for positioning the mass effectively within the rotating cone system.