Rotational Dynamics in a Pully

In summary, the system consists of a green hoop, blue solid disk pulley, and an orange sphere on a flat horizontal surface. The hoop has a mass of 2.6 kg and radius of 0.12 m, the pulley has a mass of 2.3 kg and radius of 0.09 m, and the sphere has a mass of 3.3 kg and radius of 0.23 m. The system is released from rest and the task is to find the magnitude of the linear acceleration of the hoop. Using the equations for torque and force, the acceleration of the rope over the pulley and the acceleration of the sphere can be calculated. However, the first attempt at a solution did not account
  • #1
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Homework Statement


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An green hoop with mass mh = 2.6 kg and radius Rh = 0.12 m hangs from a string that goes over a blue solid disk pulley with mass md = 2.3 kg and radius Rd = 0.09 m. The other end of the string is attached to a massless axel through the center of an orange sphere on a flat horizontal surface that rolls without slipping and has mass ms = 3.3 kg and radius Rs = 0.23 m. The system is released from rest.

What is magnitude of the linear acceleration of the hoop?

Homework Equations


T = Ia
F=ma
I_sphere = 0.0698
I_disc = 0.00932

The Attempt at a Solution


F_1 = 9.81 * mass_of_hoop = 25.506 N
F_2 = the force exerted on the sphere.
a = the acceleration of the sphere and hoop

a = ((F_1 - F_2)r_disc^2) / I_disc
a = (F_2 * r_sphere ^ 2)/I_sphere

The first equation tells is the acceleration of the rope over the pully. The second one is the acceleration of the sphere. The resultant acceleration is greater than 9.81 so I know it is wrong. I tried changing to second equation to a = ((F_2 - (m_sphere * a)) * r_sphere ^ 2)/I_sphere but that is also wrong.
 

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  • #2
use rxf=i.α for torque and Newtons equation for force and accln.
 
  • #3
What is rx.
 
  • #4
I got it. I needed to also account for the mass of the hoop in the second equation.
 
  • #5


I would suggest checking your equations and making sure they are correct for the given scenario. It seems that you are trying to use equations for linear motion to solve a rotational dynamics problem. In this scenario, the forces and accelerations are acting on different objects with different moments of inertia, so the equations need to be modified accordingly.

One approach to solving this problem could be to use the equation T = Iα for rotational motion, where T is the torque, I is the moment of inertia, and α is the angular acceleration. In this case, the torque on the pulley and the torque on the sphere are equal and opposite, so you can set them equal to each other and solve for the angular acceleration. Then, you can use the relationship between linear and angular acceleration (a = αr) to find the linear acceleration of the hoop.

Another approach could be to use the concept of conservation of energy. At the moment the system is released, the potential energy of the hoop is converted into kinetic energy as it falls and accelerates. You can set the potential energy equal to the kinetic energy and solve for the acceleration.

I would also recommend drawing a free body diagram for each object in the system and carefully considering all the forces acting on them. This can help you identify all the necessary equations and variables to solve the problem accurately.
 

What is rotational dynamics in a pulley?

Rotational dynamics in a pulley refers to the study of the rotational motion and forces involved in the movement of a pulley system, which consists of a wheel with a groove around its circumference that a rope or belt can run through.

What is the importance of understanding rotational dynamics in a pulley?

Understanding rotational dynamics in a pulley is important for designing and optimizing pulley systems, as well as analyzing and predicting their behavior. It is also essential for understanding the principles of mechanical motion and energy transfer.

What are the key components of rotational dynamics in a pulley?

The key components of rotational dynamics in a pulley include the pulley itself, the rope or belt, the load being lifted or moved, and the forces acting on the system, such as tension, friction, and gravity. The size, mass, and shape of these components can affect the rotational dynamics.

How do you calculate the mechanical advantage of a pulley system?

The mechanical advantage of a pulley system can be calculated by dividing the load being lifted by the effort force applied to the rope or belt. For example, a system with four supporting ropes will have a mechanical advantage of 4, meaning the effort force will be four times less than the load.

What are some real-world applications of rotational dynamics in a pulley?

Rotational dynamics in a pulley can be observed in various real-world applications, such as cranes, elevators, and weightlifting equipment. It is also used in simple machines, such as a flagpole or a well. Additionally, rotational dynamics in a pulley is essential in the study of rotational motion in physics and engineering.

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