A revolving door consists of four rectangular glass slabs,with the long end of each attached to a pole that acts as the rotation axis. Each slab is 2.60m tall by 1.50m wide and has mass 39.5 .
Find the rotational inertia of the entire door.
If it's rotating at one revolution every 10.5 , what's the door's kinetic energy?
The moment of inertia (rotational inertia) of the rectangular plate about the axis passing along the axis is
Iₒ=ma²/3, where a= 1.5 m.
Moment of inertia of the whole door is I=4Iₒ=4ma²/3 =4•39.5•1.5²/3=117.9 kg•m².
f=1/10.5=0.095 Hz
KE=Iω²/2=I(2πf)²/2=
=117.9•(2•π•0.095)²/2=21.1 J
To find the rotational inertia of the entire door, you need to calculate the moment of inertia for each slab of glass and then add them up.
The moment of inertia for a rectangular slab rotating around its long axis can be calculated as follows:
I = (1/12) * m * (h^2 + w^2)
Where:
I is the moment of inertia
m is the mass of the slab
h is the height of the slab
w is the width of the slab
Given that each slab has a height (h) of 2.60 m, a width (w) of 1.50 m, and a mass (m) of 39.5 kg, let's calculate the moment of inertia for each slab:
I1 = (1/12) * 39.5 * (2.60^2 + 1.50^2)
I2 = (1/12) * 39.5 * (2.60^2 + 1.50^2)
I3 = (1/12) * 39.5 * (2.60^2 + 1.50^2)
I4 = (1/12) * 39.5 * (2.60^2 + 1.50^2)
Next, add up the four moments of inertia to find the total moment of inertia for the entire door:
I_total = I1 + I2 + I3 + I4
Now, to find the kinetic energy of the revolving door, you can use the formula:
K.E. = (1/2) * I * w^2
Where:
K.E. is the kinetic energy
I is the moment of inertia
w is the angular velocity
Given that the revolving door rotates at one revolution every 10.5 s, we can convert this to angular velocity (w) in radians per second:
w = (2π) / T
Where:
w is the angular velocity
T is the period of rotation
w = (2π) / 10.5
Now, you can substitute the values of the moment of inertia (I) and the angular velocity (w) into the kinetic energy formula to find the door's kinetic energy.