A 10g marble is spun so that it rolls at 150rpm around the inside of a steel tube. The tube has a 12cm diameter. Assume the rolling resistance is small enough for the marble to maintain 150rpm for several seconds. During this time, will the marble spin in a horizontal circle or will it spiral down in the tube?
To determine whether the marble will spin in a horizontal circle or spiral down in the tube, we need to analyze the forces acting on the marble.
First, let's consider the gravitational force acting on the marble. The force due to gravity will attempt to pull the marble downwards, perpendicular to the horizontal circular motion.
Next, there is the centripetal force, which is responsible for maintaining the marble's circular motion. In this case, the centripetal force is provided by the normal force exerted by the tube on the marble. As the marble rolls inside the tube, the normal force acts radially inward, towards the center of the circle.
If the normal force provided by the tube is sufficient to counterbalance the gravitational force, the marble will spin in a horizontal circle with a constant radius. However, if the gravitational force is greater than the normal force, the marble will spiral down inside the tube.
To determine whether the marble will spiral down or spin in a horizontal circle, we can calculate the gravitational force and compare it to the centripetal force.
1. Calculate the gravitational force:
The gravitational force acting on the marble can be calculated using the formula:
F_gravity = m * g
Where:
m = mass of the marble (10g = 0.01kg)
g = acceleration due to gravity (9.8 m/s²)
F_gravity = 0.01kg * 9.8 m/s² = 0.098 N
2. Calculate the centripetal force:
The centripetal force can be calculated using the formula:
F_centripetal = m * w² * r
Where:
m = mass of the marble (0.01kg)
w = angular velocity (angular speed) in radians per second
r = radius of the circular path (tube diameter / 2)
Given:
Angular velocity (w) = 150rpm = 150 * (2π/60) rad/s
Tube diameter = 12cm = 0.12m
Radius (r) = 0.06m (half of the tube diameter)
F_centripetal = 0.01kg * (150 * (2π/60))^2 * 0.06m
Now, comparing the gravitational force (F_gravity) and the centripetal force (F_centripetal), if F_gravity is greater than F_centripetal, then the marble will spiral down in the tube. If F_centripetal is greater than or equal to F_gravity, then the marble will spin in a horizontal circle.
By evaluating these calculations, we can determine the marble's motion inside the tube.