Rolling on a Surface Which is Moving with Constant Acceleration

When an object rolls on a surface that is moving with constant acceleration, the dynamics of the rolling object are influenced by both its own rotational motion and the acceleration of the surface. To understand this topic, we need to delve into the concepts of rotational motion, Newton's laws of motion, and how they apply to systems with non-inertial frames of reference.

Understanding Rolling Motion

Rolling motion is a combination of translational motion (motion in a straight line) and rotational motion (motion around an axis). When an object rolls without slipping, the point of the object in contact with the surface is momentarily at rest relative to the surface.

The condition for rolling without slipping is given by:

$$ v = \omega r $$

where:

• ( v ) is the translational velocity of the center of mass of the rolling object.
• ( \omega ) is the angular velocity of the object.
• ( r ) is the radius of the object.

Dynamics of Rolling on an Accelerating Surface

When the surface itself is accelerating, the rolling object experiences additional forces due to the non-inertial reference frame. Let's denote the acceleration of the surface as ( a_s ).

Forces Acting on the Rolling Object

1. Gravitational Force: Acts downward, equal to ( mg ), where ( m ) is the mass of the object and ( g ) is the acceleration due to gravity.
2. Normal Force: Acts perpendicular to the surface, balancing the gravitational force in the absence of vertical acceleration.
3. Frictional Force: Acts at the point of contact, providing the torque necessary for rolling and the horizontal force to accelerate the center of mass.

Equations of Motion

The equations governing the motion of the rolling object are:

1. Translational Motion: ( F = ma ), where ( F ) is the net force acting on the object and ( a ) is the acceleration of the object's center of mass.
2. Rotational Motion: ( \tau = I\alpha ), where ( \tau ) is the net torque, ( I ) is the moment of inertia of the object, and ( \alpha ) is the angular acceleration.

Non-Inertial Reference Frame

Since the surface is accelerating, we must consider the pseudo force acting on the rolling object in the frame of the surface. This pseudo force is equal to ( -ma_s ) and acts in the opposite direction of the surface's acceleration.

Table of Differences and Important Points

Aspect	Rolling on a Stationary Surface	Rolling on an Accelerating Surface
Frame of Reference	Inertial	Non-inertial
Pseudo Force	None	( -ma_s )
Friction	Provides torque for rotation	Provides torque and compensates for surface acceleration
Acceleration of Center of Mass	Equal to ( \alpha r )	( a = \alpha r + a_s )
Net Force	( F = ma )	( F + (-ma_s) = ma )

Examples

Example 1: Rolling on an Accelerating Conveyor Belt

Consider a cylinder rolling on a conveyor belt that is accelerating to the right with an acceleration ( a_s ). The friction between the cylinder and the belt is sufficient to prevent slipping.

Given:

• Mass of the cylinder, ( m )
• Radius of the cylinder, ( r )
• Moment of inertia of the cylinder, ( I )
• Acceleration of the conveyor belt, ( a_s )

Find:

• The acceleration of the cylinder's center of mass, ( a )
• The angular acceleration of the cylinder, ( \alpha )

Solution:

1. Write down the force equation considering the pseudo force:

$$ ma_s - f = ma $$

where ( f ) is the frictional force.

1. Write down the torque equation:

$$ f \cdot r = I\alpha $$

1. Use the rolling condition ( a = \alpha r ) to combine the equations:

$$ ma_s - \frac{I}{r}a = ma $$

1. Solve for ( a ):

$$ a = \frac{ma_s}{1 + \frac{I}{mr^2}} $$

1. Find ( \alpha ) using ( a = \alpha r ):

$$ \alpha = \frac{a}{r} $$

Example 2: Rolling Sphere on an Accelerating Car

A solid sphere is rolling on the top of a car that is accelerating forward. The sphere does not slip on the surface of the car.

Given:

• Mass of the sphere, ( m )
• Radius of the sphere, ( r )
• Moment of inertia of a solid sphere, ( I = \frac{2}{5}mr^2 )
• Acceleration of the car, ( a_s )

Find:

• The acceleration of the sphere's center of mass, ( a )
• The angular acceleration of the sphere, ( \alpha )

Solution:

Follow the same steps as in Example 1, but use the moment of inertia for a solid sphere:

1. Force equation with pseudo force:

$$ ma_s - f = ma $$

1. Torque equation:

$$ f \cdot r = I\alpha $$

1. Rolling condition ( a = \alpha r ) and solve for ( a ):

$$ a = \frac{ma_s}{1 + \frac{2}{5}} = \frac{5}{7}a_s $$

1. Find ( \alpha ):

$$ \alpha = \frac{5}{7}\frac{a_s}{r} $$

In conclusion, rolling on a surface that is moving with constant acceleration involves analyzing the forces and torques in a non-inertial reference frame. The key is to account for the pseudo force due to the accelerating surface and apply the conditions for rolling without slipping to find the translational and angular accelerations of the rolling object.