Velocity of a Falling Window

In summary, the decorative wood-framed window below falls from a nominally stable vertical position, striking a person on the top of the head with a force of 1.5 kg. The window has a width of 20 cm and a height of 1.2 m, and a thickness of 15 cm. The frame is made of wood and glass, and the person is sitting 1.1 m away from the center of the base of the window.
  • #1
farshadian
2
0
Hi I'm having trouble with my injury mechanics problem where a rigid window with frame falls from a verticle position on a person sitting below.

Homework Statement



The decorative wood-framed window below has been inadvertently left
unrestrained in a nominally stable vertical position. The window pivots around its base until it strikes a person on the top of the head. Model the falling window to determine the velocity at head contact. then estimate the force applied to the head if the impulse is over 5 ms (assumed triangular contact).

Density of wood is 0.673 g/cc and glass is 2.6g/cc

DIMENSIONS are
2.0m wide frame
1.2 m high
15 cm thickness

Frame width is 20 cm everywhere and glass is in center.

a person is sitting 1.1m away from the center of the base (path of contact is


Homework Equations


I know your supposed to use impulse and moment
and the parallel axis theorm. I can't figure out how to find the CoG for the block which is one of the first steps

The Attempt at a Solution



cirumfrance of the falling obj. about the person is = pi * r /4 = 0.864 m
Area of whole thing is 2m*1.2m = 2.4 m2
Glass area is 0.8*1.6= 1.28 m2
area of the frame is 1.12m2 and its volume is 0.0064m3
volume of frame is .168 m3
mass of frame is .113 g and the window is .0166g




can anyone point me in the right direction?
 
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  • #2
welcome to pf!

hi farshadian! welcome to pf! :wink:

you need the moment of inertia of the whole window about the base (you can add the m.i. of the glass to that of the wood) …

what formulas do you know for https://www.physicsforums.com/library.php?do=view_item&itemid=31" of a rectangle?

(the area and volume don't matter)

show us what you get :smile:
 
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  • #3
Ya i figured it out,

I think you can do it without parallel axis theorem But i used it.
Ixx = Ix + mr^2 = (xy^3/3 - xy^3/3) + m*y^2 = 41.5 m^4
^^^ for wood, repeate for glass)
angular accel. (alpha) which is torque over the above inertia
torque was mgh sin (90) , so
w (omega) = root (2 alpha) = v/r .: v = 8.78 m/s

impact wise, the triangle means peak accel is at half impusle time. so you find the force at that time. but that is a different part of the question.happy pi day!
 
  • #4
farshadian said:
happy pi day!

many circular returns to you too! :biggrin:
 
  • #5


Hello,

To solve this problem, you will need to use the principles of mechanics, specifically Newton's laws of motion. The first step is to determine the acceleration of the window as it falls. This can be done using the equation F=ma, where F is the force of gravity acting on the window, m is the mass of the window, and a is the acceleration. The force of gravity can be calculated using the mass and the acceleration due to gravity (9.8 m/s^2).

Next, you will need to calculate the moment of inertia of the window. This is a measure of the window's resistance to rotational motion. The moment of inertia for a rectangular object can be calculated using the equation I = (1/12) * m * (h^2 + w^2), where m is the mass of the object, h is the height, and w is the width.

Once you have the moment of inertia, you can use the parallel axis theorem to calculate the moment of inertia about the point of contact with the person's head. The parallel axis theorem states that the moment of inertia about a point is equal to the moment of inertia about the center of mass plus the mass times the distance squared between the two points.

Finally, you can use the equation τ = Iα, where τ is the torque (force causing rotational motion), I is the moment of inertia, and α is the angular acceleration. You can solve for the angular acceleration and then use the equations v = ωr and a = αr to find the velocity and acceleration of the window at the point of contact with the person's head.

To calculate the force applied to the head, you will need to use the equation F = ma, where m is the mass of the window and a is the acceleration calculated in the previous step. You can then use the equation FΔt = mΔv to calculate the impulse, where F is the force, Δt is the time of contact (5 ms in this case), m is the mass of the window, and Δv is the change in velocity.

I hope this helps guide you in the right direction. Remember to always check your units and use consistent units throughout your calculations. Good luck!
 

What is the velocity of a falling window?

The velocity of a falling window refers to the speed at which the window is falling towards the ground. It is typically measured in meters per second (m/s).

How is the velocity of a falling window calculated?

The velocity of a falling window can be calculated using the formula v = √(2gh), where v is the velocity, g is the acceleration due to gravity (9.8 m/s²), and h is the height of the window above the ground.

Does the size or weight of the window affect its velocity when falling?

Yes, the size and weight of the window can affect its velocity when falling. A larger or heavier window will experience more air resistance, which can slow down its velocity. However, this effect is typically minimal for windows of typical sizes and weights.

Can the velocity of a falling window be increased or decreased?

Yes, the velocity of a falling window can be increased or decreased by changing the height from which it is dropped, or by changing its mass. For example, dropping a window from a higher height will result in a higher velocity, while reducing its mass will result in a lower velocity.

Is the velocity of a falling window affected by air resistance?

Yes, the velocity of a falling window is affected by air resistance. As the window falls, it will experience air resistance, which will gradually slow down its velocity. This effect becomes more significant as the window's velocity increases.

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