Calculating moment of inertia about a door hinge.à

In summary, a solid door with a mass of 39.30 kg and dimensions of 2.34 m x 1.68 m x 3.23 cm has a moment of inertia of 37 kgm^2 about an axis through its hinges. To find the rotational kinetic energy, the tangential speed of the edge must be converted to angular speed by multiplying it by the width of the door. This results in a rotational kinetic energy of 30.6 J.
  • #1
Becca93
84
1
Homework Statement
A solid door of mass 39.30 kg is 2.34 m high, 1.68 m wide, and 3.23 cm thick.

What is the moment of inertia of the door about the axis through its hinges?

If the edge of the door has a tangential speed of 76.5 cm/s, what is the rotational kinetic energy of the door?


The attempt at a solution

I don't really know where to start. Should I find the center of mass of the door? Because I = Ʃmr^2?

I know how to solve for the second half (Ek = (1/2)Iω^2), but I'm not sure how to calculate I.
 
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  • #2
The moment of inertia of a thin rod of mass m and length L about an axis at one of its ends and perpendicular to the rod is given by
I = [itex]\frac{mL^{2}}{3}[/itex]

Now let us imagine the door to be a set of thin rods parallel to the axis passing through the hinges.
 
  • #3
grzz said:
The moment of inertia of a thin rod of mass m and length L about an axis at one of its ends and perpendicular to the rod is given by
I = [itex]\frac{mL^{2}}{3}[/itex]

Now let us imagine the door to be a set of thin rods parallel to the axis passing through the hinges.

I have the answer to the first question, but I'm not nearly as well off with the second question as I had assumed.

I got 37 kgm^2 for the first question.

I assumed you would take that value and take 76.5 cm/s (.765 m/s) as ω and plug it into
E = (1/2)Iω^2

I get 10.8 when I do that (I used m/s and I'm assuming the units are joules), but that's incorrect.

Any advice?
 
  • #4
[itex]\omega[/itex] is the ANGULAR speed i.e. measured in ANGLE per second i.e. rad/s and so you have the convert the TANGENTIAL speed given to ANGULAR speed.
 
  • #5
grzz said:
[itex]\omega[/itex] is the ANGULAR speed i.e. measured in ANGLE per second i.e. rad/s and so you have the convert the TANGENTIAL speed given to ANGULAR speed.

Oh! Okay. How do I calculate angular speed from tangential speed?
 
  • #6
angle (in radians) rotated = distance along the arc / radius
therefore distance along arc = radius x angle
i.e. distance along arc/time = radius x (angle /time)
v = r x w
v in m/s because linear or tangential
w in rad/s because angular
 
  • #7
grzz said:
angle (in radians) rotated = distance along the arc / radius
therefore distance along arc = radius x angle
i.e. distance along arc/time = radius x (angle /time)
v = r x w
v in m/s because linear or tangential
w in rad/s because angular

So to get velocity, I take the linear velocity and multiply it by the width of the door to get ω, and then plug that value into the energy equation?

Edit: when I do that I get 30.6 J and that is incorrect.
 
  • #8
Becca93 said:
... I take the linear velocity and multiply it by the width of the door to get ω ...

v = R[itex]\omega[/itex]
therefore ... = [itex]\omega[/itex]
 
  • #9
grzz said:
v = R[itex]\omega[/itex]
therefore ... = [itex]\omega[/itex]

Got it! Thanks!
 

FAQ: Calculating moment of inertia about a door hinge.à

1. How do you calculate the moment of inertia about a door hinge?

The moment of inertia about a door hinge can be calculated by using the formula: I = m x r^2, where I is the moment of inertia, m is the mass of the door, and r is the distance from the door hinge to the center of mass of the door.

2. What is the importance of calculating the moment of inertia about a door hinge?

Calculating the moment of inertia about a door hinge is important because it helps determine the resistance of the door to rotational motion. This information is essential for understanding the stability and strength of the door, as well as for designing hinges and other door components.

3. How does the shape of the door affect the moment of inertia about a door hinge?

The shape of the door can greatly affect the moment of inertia about a door hinge. A door with a larger mass and farther center of mass from the hinge will have a higher moment of inertia, making it more resistant to rotational motion. Additionally, the distribution of mass along the door can also impact the moment of inertia.

4. Can the moment of inertia about a door hinge change?

Yes, the moment of inertia about a door hinge can change if there are alterations made to the door, such as adding or removing weight, changing the shape, or adjusting the position of the center of mass. Additionally, the moment of inertia can also be affected by external forces, such as wind or people pushing or pulling on the door.

5. How is the moment of inertia about a door hinge used in engineering and design?

The moment of inertia about a door hinge is an important factor in engineering and design as it helps determine the strength and stability of the door. Engineers and designers can use this information to select appropriate hinges and materials, as well as to ensure that the door can withstand expected forces and movements without breaking or becoming unstable.

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