Some dragonflies splash down onto the surface of a lake to clean themselves. After this dunking, the dragonflies gain altitude, and then spin rapidly at about 1100 rpm to spray the water off their bodies. When the dragonflies do this "spin-dry," they tuck themselves into a "ball" with a moment of inertia of 2.1×10−7kg⋅m2.
How much energy must the dragonfly generate to spin itself at this rate?
To find the energy required for the dragonfly to spin itself at a rate of 1100 rpm, we need to calculate its rotational kinetic energy.
The formula for rotational kinetic energy is:
Rotational Kinetic Energy = (1/2) * moment of inertia * angular velocity^2
Given:
Moment of inertia (I) = 2.1 × 10^(-7) kg⋅m^2
Angular velocity (ω) = 1100 rpm = (1100/60) rev/s
First, let's convert the angular velocity from revolutions per minute (rpm) to radians per second (rad/s):
Angular velocity (ω) = (1100/60) * (2π rad/1 rev)
Angular velocity (ω) ≈ 114.62 rad/s
Now we can substitute these values into the formula:
Rotational Kinetic Energy = (1/2) * (2.1 × 10^(-7) kg⋅m^2) * (114.62 rad/s)^2
Simplifying the equation:
Rotational Kinetic Energy = (1/2) * (2.1 × 10^(-7) kg⋅m^2) * 13128.7444 rad^2/s^2
Finally, calculating the value:
Rotational Kinetic Energy ≈ 1.44 × 10^(-2) Joules
Therefore, the dragonfly needs to generate approximately 1.44 × 10^(-2) Joules of energy to spin itself at a rate of 1100 rpm.
To calculate the energy required to spin the dragonfly at a given rate, we need to use the formula for rotational kinetic energy.
The formula for rotational kinetic energy is:
KE(rot) = (1/2) * I * ω^2
Where:
KE(rot) is the rotational kinetic energy,
I is the moment of inertia, and
ω is the angular velocity in radians per second.
Given values:
I = 2.1×10^(-7) kg⋅m^2 (moment of inertia)
ω = 1100 rpm = 1100 * (2π/60) rad/s (angular velocity, converted from rpm to rad/s)
Let's plug in the values into the formula and calculate the energy:
KE(rot) = (1/2) * 2.1×10^(-7) kg⋅m^2 * (1100 * (2π/60) rad/s)^2
First, let's simplify the angular velocity:
ω = 1100 * (2π/60) rad/s
ω = 114.803 rad/s (approx.)
Now we can calculate the energy:
KE(rot) = (1/2) * 2.1×10^(-7) kg⋅m^2 * (114.803 rad/s)^2
KE(rot) = 0.5 * 2.1×10^(-7) kg⋅m^2 * (13170.55 rad^2/s^2)
KE(rot) = 1.38479105×10^(-3) kg⋅m^2⋅rad^2/s^2
Therefore, the energy required for the dragonfly to spin itself at this rate is approximately 1.38479105×10^(-3) kg⋅m^2⋅rad^2/s^2.
KE in the rotation= 1/2 *momnet*w^2
moment is given
w=2PI*1100/60 rad/sec
solve for KE