Max Ramp Angle for Rolling Ball: mu & Theta

In summary, the conversation discusses the maximum angle theta that a ball can roll down a ramp without slipping when released from rest, with a coefficient of static friction of mu. The suggested formula for theta is inverse(tan(mu)) but the book's answer uses a coefficient of 3.5 times mu. The conversation also raises a question about two balls rolling down a ramp, one filled with fluid and one without, and which one would reach the bottom first. The conclusion is that the fluid-filled ball may roll slightly slower due to friction and viscosity, but the difference would be negligible. The conversation also mentions the role of air friction in the problem, but it is determined to not be a significant factor.
  • #1
rhuala
3
0
Ball mass m and radius r rolls down ramp with coefficient of static friction of mu. If the ball is released from rest what is the maximum angle theta of the ramp that the ball rolls without slipping?

I've got theta max = inverse(tan(mu)) but the answer in the book is

theta = inverse(tan(3.5mu))

I'm not sure where the 3.5 comes in could someone please explain?

Also if 2 balls roll down a ramp, one is filled with fluid the other not which one reaches the bottom first. The mass cancels out the the equation for the acceleration so it seems to me they should reach the bottom at the same time. Just wanted to verify that this is correct.

Thanks in advance

Carla
 
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  • #2
"Also if 2 balls roll down a ramp, one is filled with fluid the other not which one reaches the bottom first. "

Does the fluid-filled one weigh more?

It seems to me that the lighter ball would reach the ground first.
1] The lighter ball will alternate between rolling and freefalling (after each little bump)
2] The The heavier ball also resists the turning caused by friction, thus will accelerate slower.
 
  • #3
rhuala said:
I've got theta max = inverse(tan(mu)) but the answer in the book is

theta = inverse(tan(3.5mu))

I'm not sure where the 3.5 comes in could someone please explain?
Perhaps you are confusing this problem with finding the maximum angle that an incline can be increased before an object begins sliding down?

Hint: There is a net force acting down the incline---apply Newton's 2nd law for both translation and rotation.
 
  • #4
Friction and viscosity of the fluid should cause the fluid filled ball to roll slightly slower.
 
  • #5
Friction and viscosity of the fluid should cause the fluid filled ball to roll slightly slower.

Can you (or someone) show this through formula please, I'm not sure this is correct...
 
  • #6
I don't know the formula's for this, but here's an similar example.

Replace the fluid with an object of a certain mass with a low coefficient of friction. As the ball rolls, the object is raised a bit, and then starts sliding inside of the ball as the ball rolls. In addition to increasing the kinetic engergy of the ball, temperature energy is also being increased (from the friction).

Since the initial potential engery is the same for both balls at the start, the ball with the increasing temperature energy ends up with less of an increase in kinetic energy.
 
  • #7
ignore what jeff said... air friction play no role in this problem
identify the problem and read #3 post carefully... let me know how far you get or where you stuck b4 i can further help you... this problem is not as hard as you think... and don't expect it is a easy problem... (am i contradicting myself?) :wink:
 
  • #8
ignore what jeff said... air friction play no role in this problem
I never mentioned air friction. One of the questions concerened fluid in a ball, which is a source of internal friction (viscosity).
 

1. What is the significance of the maximum ramp angle for a rolling ball?

The maximum ramp angle for a rolling ball, also known as the critical angle, is the steepest angle at which a ball can roll without slipping. It is an important factor to consider in designing ramps or inclines for objects that roll, such as cars or billiard balls, to prevent slipping and ensure safe movement.

2. How is the maximum ramp angle determined?

The maximum ramp angle is determined by the coefficient of friction (mu) between the rolling object and the surface it is rolling on, as well as the angle of inclination (theta) of the ramp. It can be calculated using the formula tan(theta) = mu, with the resulting value being the critical angle.

3. What factors affect the maximum ramp angle?

There are several factors that can affect the maximum ramp angle for a rolling ball. These include the coefficient of friction, the weight and shape of the rolling object, the surface texture and material of the ramp, and any external forces acting on the object.

4. Can the maximum ramp angle be exceeded?

In theory, the maximum ramp angle should not be exceeded as it can lead to slipping and loss of control of the rolling object. However, in certain circumstances, such as with high-quality tires or specialized equipment, the critical angle may be slightly exceeded without significant consequences.

5. How can the maximum ramp angle be increased?

The maximum ramp angle can be increased by increasing the coefficient of friction between the rolling object and the surface it is rolling on. This can be achieved by using materials with higher friction properties, such as rubber or sandpaper, or by adding external means of increasing friction, such as sand or adhesive strips, to the surface of the ramp.

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