What Is the Angle of Incline for a Sphere Rolling Without Slipping?

In summary: So you can solve for either one of them using the other equation as a hint. In summary, the sphere rolls down an incline without slipping, and the linear acceleration of the center of mass is 0.22g. The angle the incline makes with the horizontal is the resultant of the translational and rotational accelerations.
  • #1
hejo
4
0

Homework Statement



A uniform solid sphere rolls down an incline without slipping. If the linear acceleration of the center of mass of the sphere is 0.22g, then what is the angle the incline makes with the horizontal?

Homework Equations



a=gsin(theta)

The Attempt at a Solution



The only thing I can think of is acceleration down an incline=gsin(theta). Since we have a=0.22g, I made gsin(theta)=0.22g and thereby arcsin(0.22) should give me theta...but it doesn't...any help please??
 
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  • #2
hejo said:
The only thing I can think of is acceleration down an incline=gsin(theta).
That's the acceleration down an incline if the only force acting parallel to the incline is gravity, for example when a box slides down a frictionless incline. Hint: Since the sphere rolls without slipping, the incline cannot be frictionless. (Solve for the acceleration using Newton's 2nd law twice; once for translation, once for rotation.)
 
  • #3
Okay. I set up my equation for translational motion to be ma=mgsin(theta) - (coefficient of friction)*mgcos(theta). So a=gsin(theta) - gcos(theta)*(coefficient of friction).

Is that right? I don't understand how I can get acceleration from rotational motion.
 
  • #4
hejo said:
Okay. I set up my equation for translational motion to be ma=mgsin(theta) - (coefficient of friction)*mgcos(theta). So a=gsin(theta) - gcos(theta)*(coefficient of friction).

Is that right?
No, it's not quite right. Since there's no slipping, the relevant friction is static friction. And static friction can be anything up to a maximum of μN. So you can't just set it equal to that maximum value. Just call the friction "F" and continue.
I don't understand how I can get acceleration from rotational motion.
That friction force also exerts a torque on the sphere which produces an angular acceleration. Set up an equation for that. (You can relate the angular acceleration to the translational acceleration.)

Note that you have two unknowns--the friction and the acceleration--but you also have two equations.
 

Related to What Is the Angle of Incline for a Sphere Rolling Without Slipping?

1. How does the angle of incline affect rotation down an incline?

The angle of incline has a significant impact on the rotation of an object down an incline. As the angle increases, the object will experience a greater acceleration and thus rotate faster down the incline. This is because a steeper incline increases the component of the object's weight that acts parallel to the surface, resulting in a greater force and acceleration.

2. What factors contribute to the rotational motion of an object down an incline?

There are several factors that contribute to the rotational motion of an object down an incline, including the object's mass, shape, and distribution of mass, as well as the angle of incline and any external forces acting on the object. The presence of friction between the object and the incline can also affect the rotational motion.

3. How does friction affect rotation down an incline?

Friction plays a role in the rotational motion of an object down an incline by acting in the opposite direction of the object's motion. This results in a torque that slows down the rotation of the object. The amount of friction present depends on the coefficient of friction between the object and the incline, as well as the normal force acting on the object.

4. Can an object rotate down an incline without any external forces?

No, an object cannot rotate down an incline without any external forces. This is because an external force, such as the force of gravity, is needed to cause the object to accelerate and rotate down the incline. In the absence of friction, the object would continue to rotate at a constant speed once it reaches the bottom of the incline.

5. How is rotational motion down an incline related to linear motion?

Rotational motion down an incline is closely related to linear motion. As the object rotates, it also moves linearly down the incline. The relationship between the two can be described using the principles of torque, angular acceleration, and linear acceleration. The rotational motion can also be converted to linear motion using the radius of the object and its angular velocity.

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