Rotational Motion of a cylinder

In summary, the conversation discusses a problem involving a hollow cylinder rolling on a horizontal surface and then reaching a 15 degree incline. The questions asked are: (a) how far up the incline will it go? and (b) how long will it be on the incline before it arrives back at the bottom? The equations used to solve the problem are conservation of energy, SOH CAH TOA, and kinematics. The solution for (a) is 7.3 m and for (b) it is suggested to use simple kinematic equations to calculate the time taken to go up the incline and then doubling it to get the desired result. The important part is to figure out the acceleration.
  • #1
Carpe Mori
19
0

Homework Statement


A hollow cylinder (hoops) is rolling on a horizontal surface at speed v = 4.3 m/s when it reaches a 15 degree incline (a) how far up the incline will it go? (b) how long will it be on the incline before it arrives back at the bottom?


Homework Equations



PE = KE + RE
SOH CAH TOA

w = v/r

I = mr^2


The Attempt at a Solution



A) I figured out letter A with conservation of Energy (yawn) distance = 7.3 m

B)I am not sure exactly how to solve this problem. If it was a non rotating cube i could do it very easilly =D. I am sure the fact that it is rotating though affects it somehow. For a cube I would use the Kinetics equations (y = -.5gt^2 + vt + y0) to solve for time. Could someone help me out thanks =D
 
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  • #2
the time taken to go up the incline=the time taken to come down.
calculate time taken to go up sing simple kinamtic relations and then double it to get the desired result.

use velocity final=0
velocity[initial]=4.3 m/s
distance=7.3 m

now the important part is accln- i leave it to you to figure it out.
use =s=ut+1/2at^2
 
  • #3

I would like to point out that rotational motion of a cylinder involves both linear and angular motion. In this case, the cylinder is rolling on a horizontal surface, which means it has both translational and rotational kinetic energy. When it reaches the incline, it will have potential energy due to its height and kinetic energy due to its rotational and translational motion.

To solve part (b) of the problem, we can use the conservation of energy equation, as you did for part (a). However, since the cylinder is now on an incline, we need to consider the change in its potential energy as well as its kinetic energy. We can use the same equation, PE = KE + RE, but we need to include the change in potential energy due to the incline.

Using the conservation of energy equation, we can set the initial kinetic energy (KEi) equal to the final potential energy (PEf) and kinetic energy (KEf). This will give us the equation:

KEi = PEf + KEf

Since we know the initial kinetic energy (KEi) and the final potential energy (PEf) is zero at the bottom of the incline, we can solve for the final kinetic energy (KEf).

KEf = KEi - PEf

We can also use the rotational kinetic energy equation, KE = 1/2Iw^2, where I is the moment of inertia and w is the angular velocity. Since the cylinder is rolling without slipping, we can relate the linear velocity (v) and angular velocity (w) using the equation w = v/r, where r is the radius of the cylinder.

Substituting this into the rotational kinetic energy equation, we get:

KEf = 1/2I(v/r)^2

We can then equate this to the equation for final kinetic energy that we obtained from the conservation of energy equation.

KEi - PEf = 1/2I(v/r)^2

Solving for time, we get:

t = √(2I(v/r)^2/KEi)

Where I is the moment of inertia of the cylinder, which can be calculated using the formula I = mr^2, where m is the mass of the cylinder and r is the radius.

I hope this helps you solve part (b) of the problem. As a scientist, it is important to consider all aspects of a problem and use relevant equations
 

What is rotational motion?

Rotational motion is the movement of an object around an axis or center point. This type of motion is often seen in objects that spin or rotate, such as a cylinder.

What is a cylinder?

A cylinder is a three-dimensional object with a circular base and straight sides. It can be thought of as a stack of circles, with each circle getting smaller as they stack upwards.

How does rotational motion affect a cylinder?

When a cylinder experiences rotational motion, it rotates around its central axis. This causes the cylinder to have a specific angular velocity, which is the rate at which it rotates. The rotational motion also affects the distribution of mass and the moment of inertia of the cylinder.

What is the moment of inertia?

The moment of inertia is a property of an object that describes how difficult it is to change the object's rotational motion. It is influenced by the distribution of mass and the shape of the object. In the case of a cylinder, the moment of inertia is affected by the radius and length of the cylinder.

How is rotational motion of a cylinder calculated?

The rotational motion of a cylinder can be calculated using the equation θ = ωt, where θ is the angular displacement (in radians), ω is the angular velocity (in radians per second), and t is the time (in seconds). This equation can also be used to calculate the angular acceleration of the cylinder.

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