Linear acceleration and moments of intertia

[yugo2113]

Im having problems with two questions, the first is linear acceleration, A 6.00-kg cylindrical reel with a radius of 0.600 m and a frictionless axle, starts from rest and speeds up uniformly as a 4.00-kg bucket falls into a well, making a light rope unwind from the reel. The bucket starts from rest and falls for 4.00 s. Find the linear acceleration of the falling bucket.

The formula I am using which I am not sure if it is correct is tmg/mt(1/2)M where t is time, m is the mass of the bucket, g is gravity, and M is the mass of the reel.

attempting to solve this problem I always end up with 3.26, somewhere I think I have my math or variables mixed up and can't come to the right conclusion


the second problem I am having trouble with is moments of interia, Four objects are held in position at the corners of a rectangle by light rods with M1=2.70kg and located at (-2m,3m), M2=1.70kg and located at (2m,3m), M3=4.10kg and located at (2m,-3m) and M4=2.10kg and located at (-2m,-3m). M stands for the masses of the objects. Find the moments of intertia along the x-axis and the y-axis respectively.

I found the moment of intertia of the system at the origin of (0m,0m) by adding the x and y coordinates and squaring them to get 9 for the distance of each mass and then multiplying that by each mass and adding each individual mass together to get the answer of 137.8kg/m^2 however that formula doesn't seem to work when I try to find the x-axis or y-axis moment of intertia by multiplying the radius of each mass by its respective mass then adding all 4 masses together. please help.

 

[alphysicist]

Hi yugo2113,

Im having problems with two questions, the first is linear acceleration, A 6.00-kg cylindrical reel with a radius of 0.600 m and a frictionless axle, starts from rest and speeds up uniformly as a 4.00-kg bucket falls into a well, making a light rope unwind from the reel. The bucket starts from rest and falls for 4.00 s. Find the linear acceleration of the falling bucket.

The formula I am using which I am not sure if it is correct is tmg/mt(1/2)M where t is time, m is the mass of the bucket, g is gravity, and M is the mass of the reel.

This formula is not correct. If you notice, it doesn't have the right units (m/(kg s^2) ).

However, the formula is very close to being right. After the time cancels (which is good since the acceleration does not depend on time), it looks to me like you might have just made a simple algebra error. Try looking over your work again, and if you can't find the error please post your work.

attempting to solve this problem I always end up with 3.26, somewhere I think I have my math or variables mixed up and can't come to the right conclusion


the second problem I am having trouble with is moments of interia, Four objects are held in position at the corners of a rectangle by light rods with M1=2.70kg and located at (-2m,3m), M2=1.70kg and located at (2m,3m), M3=4.10kg and located at (2m,-3m) and M4=2.10kg and located at (-2m,-3m). M stands for the masses of the objects. Find the moments of intertia along the x-axis and the y-axis respectively.

I found the moment of intertia of the system at the origin of (0m,0m) by adding the x and y coordinates and squaring them to get 9 for the distance of each mass and then multiplying that by each mass and adding each individual mass together to get the answer of 137.8kg/m^2 however that formula doesn't seem to work when I try to find the x-axis or y-axis moment of intertia by multiplying the radius of each mass by its respective mass then adding all 4 masses together. please help.

I don't think I follow what you've done. If you're trying to find the moment of inertia about the x-axis, you need the distance r that each mass is from the x-axis. What is that distance for each mass? Do you get the right answer for the moment of inertia?

 

[baba89]

In order to solve the first problem, the key is to understand that the reel and the bucket are connected by a light rope and therefore, the acceleration of the bucket will also be the acceleration of the reel. This means that we can use the equation F=ma, where F is the force exerted by the bucket on the reel, m is the mass of the reel and a is the acceleration of the reel. Since the reel is initially at rest, the force F will be equal to the weight of the bucket, which is given by mg. Therefore, we can rewrite the equation as mg=ma, and solving for a, we get a=g=9.8 m/s^2. This means that the acceleration of the falling bucket and the reel is 9.8 m/s^2.

For the second problem, the moment of inertia is a measure of an object's resistance to rotational motion. In order to find the moment of inertia along the x-axis and y-axis, we need to use the formula I=Σmr^2, where I is the moment of inertia, m is the mass of the object and r is the distance from the axis of rotation. For the x-axis, we can calculate the moment of inertia by adding the individual moments of inertia of each object. For example, for M1, we have I1=m1r1^2=(2.70 kg)(2m)^2=10.8 kg/m^2. Similarly, we can calculate the moments of inertia for M2, M3, and M4 and then add them together to get the total moment of inertia along the x-axis. The same process can be followed for the y-axis. The reason why your initial formula of adding the x and y coordinates and squaring them did not work is because it does not take into account the mass of each object. I hope this helps to clarify the concepts of linear acceleration and moments of inertia.