Rotational Equilibrium and dynamics

In summary: I meant to write "In summary, the conversation discusses two problems related to physics. The first problem involves a hoop rolling without slipping and asks for the distance it travels up an incline. The second problem involves a bucket on a rope and asks for the minimum force needed to raise it using a crank. Both problems require the use of equations and conservation of energy to solve."
  • #1
ryomaechizen
3
0
Please help me with these problems:

1) In a circus performance, a large 4.0 kg hoop with a radius of 2.0 m rolls without slipping. If the hoop is given an angeular speed of 6.0 rad/s while rolling on the horizontal and is allowed to roll up a ramp inclined at 15 degrees with the horizontal, how far (measured along the incline) does the hoop roll?

2) A wooden bucket filled with water has a mass of 75 kg and is attached to a rope that is wound around a cylinder with a radius of 0.075 m. A crank with a turning radius of .25 m is attached to the end of the cylinder. Wht minimum force directed perpendicularly to the crank handle is required to raise the bucket?

Thanks in advance.
 
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  • #2
Show your work and you'll get plenty of help.

Hints: For #1, use conservation of energy; for #2, consider the torques exerted on the cylinder.
 
  • #3
For the first problem, we use the equation:
.5mv^2 + .5Iw^2 + mgh

right?

I set it up as (Mechanical energy initial = Mechanical energy final)
and I was wondering if this is the right way to set it up. For some reason, I can't get the right answer.
 
  • #4
I think you equation should be 1/2mv^2 + 1/2Iw = mgh beacuse all the initial linear kinetic energy and rotational kinetic energy will be converted into gravitational potential energy as the hoop travels up the incline.
 
  • #5
ryomaechizen said:
For the first problem, we use the equation:
.5mv^2 + .5Iw^2 + mgh

right?
Right.
I set it up as (Mechanical energy initial = Mechanical energy final)
and I was wondering if this is the right way to set it up. For some reason, I can't get the right answer.
That's the right approach. Show what you did and we'll take a look. Two things to pay attention to: (1) how v relates to w, and (2) how distance up the incline relates to the change in height of the hoop's center of mass.
 
  • #6
I apologise for confusing the matter, but cud you please explain why the equation is +mgh and not =mgh
 
  • #7
I think he had written the total energy of the hoop.

Total Energy = Translational KE + Rotational KE + Grav. Potantial Energy.

= .5mv^2 + .5Iw^2 + mgh

which is the conserved quantity.
 
  • #8
Sorry, my mistake.
 

1. What is rotational equilibrium?

Rotational equilibrium refers to a state in which an object is not rotating or is rotating at a constant rate without any changes in its angular velocity. In other words, there is no net torque acting on the object, causing it to maintain its current rotational state.

2. How is rotational equilibrium different from translational equilibrium?

Rotational equilibrium deals with the balance of torques on an object, while translational equilibrium deals with the balance of forces on an object. In rotational equilibrium, the object may still be moving, but its rotational motion is constant. In translational equilibrium, the object is not moving at all.

3. What factors affect rotational equilibrium?

The factors that affect rotational equilibrium include the distribution of mass in an object, the distance of the force from the axis of rotation, and the magnitude of the force. These factors can determine the amount of torque acting on an object and whether it will be in rotational equilibrium.

4. How can you determine if an object is in rotational equilibrium?

An object is in rotational equilibrium if the net torque acting on it is zero. This can be determined by calculating the sum of all torques acting on the object and making sure it equals zero. If the object is not rotating or is rotating at a constant rate, it is also in rotational equilibrium.

5. How does angular momentum relate to rotational equilibrium?

Angular momentum is a measure of an object's tendency to continue rotating at a constant rate. In rotational equilibrium, the angular momentum of an object will remain constant as there is no net torque to change its rotational motion. If an external torque is applied, the angular momentum will change, causing the object to accelerate or decelerate in its rotation.

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