Rotational kinematics/dynamics problem

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In summary, the problem involves a plank with a mass of 6 kg being pulled by a constant force of 6 N, while riding on top of two identical rollers with a mass of 2 kg each. The rollers roll without slipping on a horizontal surface and the plank does not slip on the rollers. The goal is to calculate the linear acceleration of the plank and rollers, as well as the magnitudes of the friction forces acting between the plank and rollers, and between the rollers and the surface. This can be done by drawing free body diagrams and considering the relationships between the distances and motion of the objects. Several equations can be used to solve for the forces and accelerations.
  • #1
pppeachykeen
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Homework Statement



A plank with mass M = 6 kg rides on top of two identical solid cylindrical rollers that have R = 0.05 m and m = 2 kg. The plank is pulled by a constant horizontal force F of magnitude 6 N applied to an endpoint of the plank and perpendicular to the axes of both cylinders. The cylinders roll without slipping on a horizontal surface and the plank does not slip on the cylinders either. Knowing that there must be rolling friction (not sliding friction) between the plank and the cylinders and between the cylinders and the surace, calculate:

(a)the linear acceleration of the plank and of the rollers
(b)the magnitudes of all friction forces that are acting.
http://img389.imageshack.us/img389/61/figureio5.th.jpg

Homework Equations


Not sure...

The Attempt at a Solution


I have no idea how to go about this...any help with starting would be appreciated.
 
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  • #2
Draw free body diagrams for each cylinder and one for the plank. Each cylinder has a horizontal frictional force acting at the top and bottom. The plank has three horizontal forces acting. All vertical forces are balanced and need not be considered (although that is an interesting problem too). Consider the relationship between the distances moved by the plank and the rollers, and the relationships between the angular and linear motion of the rollers.

You will have several equations relating the applied forces to the accelerations that can be solved simultaneously to find the forces and accelerations.
 
  • #3


As a scientist, it is important to approach problems like this with a systematic and analytical mindset. We can start by identifying the given information and the unknowns. From the given information, we know the masses and dimensions of the plank and rollers, the applied force, and the type of motion (rolling without slipping). The unknowns are the linear accelerations of the plank and rollers, and the magnitudes of the friction forces.

Next, we can use the principles of rotational kinematics and dynamics to analyze the problem. We know that the plank and rollers will experience both translational and rotational motion due to the applied force. The plank will have a linear acceleration, while the rollers will have both a linear and angular acceleration. We can use the equations for linear and angular motion, along with the given information, to solve for the unknowns.

To calculate the linear acceleration of the plank, we can use Newton's Second Law, which states that the sum of the forces acting on an object is equal to its mass times its acceleration. In this case, the only horizontal force acting on the plank is the applied force F. We can set up the equation as follows:

ΣF = ma

F - Ff = ma

Where Ff is the friction force between the plank and the rollers. To find the value of Ff, we can use the equation for rolling friction, which states that the friction force is equal to the coefficient of rolling friction (μ) times the normal force (N) between the two surfaces. In this case, the normal force is equal to the weight of the plank, which is given by mg. So we have:

Ff = μmg

Substituting this into our equation, we get:

F - μmg = ma

We can rearrange this to solve for a:

a = (F - μmg)/m

Using the given values, we can plug in the numbers and solve for a. This will give us the linear acceleration of the plank.

To calculate the linear acceleration of the rollers, we can use the same equation, but this time we need to consider the forces acting on the rollers. In addition to the applied force and the friction force, the rollers will also experience a normal force from the surface. This normal force will be equal to the weight of the rollers, given by mg. So our equation becomes:

ΣF = ma

Ff - F - mg = ma

We can rearrange this
 

1. What is rotational kinematics/dynamics?

Rotational kinematics/dynamics is the study of the motion of objects that are rotating or moving in a circular path. It involves understanding concepts such as angular velocity, angular acceleration, torque, and moment of inertia.

2. How is rotational kinematics/dynamics different from linear kinematics/dynamics?

Rotational kinematics/dynamics deals with the motion of objects in circular or curved paths, while linear kinematics/dynamics deals with the motion of objects in straight lines. Additionally, rotational kinematics/dynamics involves angular quantities such as velocity and acceleration, while linear kinematics/dynamics involves linear quantities.

3. What is the difference between angular velocity and linear velocity?

Angular velocity refers to the rate at which an object is rotating, while linear velocity refers to the rate at which an object is moving in a straight line. Angular velocity is measured in radians per second, while linear velocity is measured in meters per second.

4. How do you calculate torque in a rotational kinematics/dynamics problem?

Torque is calculated by multiplying the magnitude of the force applied to an object by the perpendicular distance from the axis of rotation to the point where the force is applied. The direction of torque is determined by the right-hand rule, with the thumb pointing in the direction of the force and the fingers curling in the direction of rotation.

5. How does moment of inertia affect rotational kinematics/dynamics?

Moment of inertia is a measure of an object's resistance to rotational motion. Objects with a larger moment of inertia require more torque to rotate, while objects with a smaller moment of inertia require less torque. It is dependent on the mass and distribution of mass in an object.

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