A lawn mower has a flat, rod-shaped steel blade that rotates about its center. The mass of the blade is 0.65kg and its length is 0.55m. What is the rotational energy of the blade at its operating angular speed of 3450 rpm? If all of the rotational kinetic energy of the blade could be converted to gravitational potential energy, to what height would the blade rise?
Use the rotational energy equation, E = 0.5Iω², where I is the moment of inertia for a long thin rod with an axis through it's midpoint (I = (1/12)mL²).
E = 0.5Iω²
E = 0.5 [1/12)mL²] ω²
E = (1/24) mL² ω²
Find ω by multiplying the rpm by 2π and then dividing by 60 to put units into rad/sec (or multiply ω by π/30)
3450 * 2π = 21 676.9... rad/min
(21 676.9... rad/min)/60 = 361.28... rad/sec
Insert into energy eqn and solve
E = (1/24) mL² (361.28...rad/s)²
E = (1/24)(0.65kg)(0.55m²)(361.28...rad/s)²
E = ~1069 J
Well, isn't this grass-cutting gadget quite the performer! Let's calculate its rotational energy and the potential height it could aspire to!
To find the rotational energy of the blade, we can use the formula:
Rotational Energy = (1/2) * Moment of Inertia * Angular Speed^2,
where the Moment of Inertia is determined by the shape of the object. For a rod rotating about its center:
Moment of Inertia = (1/12) * Mass * Length^2.
Plugging in the values, we have:
Moment of Inertia = (1/12) * (0.65 kg) * (0.55 m)^2.
Calculating this gives us the Moment of Inertia. Now, let's convert the operating angular speed of 3450 rpm to radians per second. One complete rotation is 2π radians, and there are 60 seconds in a minute, so the conversion is:
Angular Speed = (3450 rpm) * (2π rad/min) * (1 min/60 s).
Now that we have all the necessary data, let's calculate the rotational energy:
Rotational Energy = (1/2) * Moment of Inertia * Angular Speed^2.
Now, to address the second part of the question - the potential height the blade could reach if all of its rotational kinetic energy were converted to gravitational potential energy:
Gravitational Potential Energy = Mass * Gravitational Acceleration * Height.
Since the only energy being transferred is the rotational kinetic energy, we can equate it to the gravitational potential energy:
Rotational Energy = Mass * Gravitational Acceleration * Height.
Now, we can solve for the height:
Height = Rotational Energy / (Mass * Gravitational Acceleration).
With these calculations, you should be able to find the rotational energy and the height the blade would rise to. Happy cutting!
To calculate the rotational energy of the blade, we can use the formula:
Rotational energy = 0.5 * I * ω^2
Where:
- I is the moment of inertia of the blade
- ω is the angular velocity of the blade
The moment of inertia of a rod rotating about its center is given by the formula:
I = (1/12) * m * L^2
Where:
- m is the mass of the blade
- L is the length of the blade
Let's start by calculating the moment of inertia (I) of the blade:
I = (1/12) * m * L^2
= (1/12) * 0.65 kg * (0.55 m)^2
= 0.0319 kg·m²
Next, let's convert the angular velocity from rpm (revolutions per minute) to rad/s (radians per second):
angular velocity (ω) = 2π * rpm / 60
= 2π * 3450 / 60 rad/s
= 360.88 rad/s
Now, let's substitute the values into the rotational energy formula:
Rotational energy = 0.5 * I * ω^2
= 0.5 * 0.0319 kg·m² * (360.88 rad/s)^2
= 2063.21 J
Therefore, the rotational energy of the blade at its operating angular speed is approximately 2063.21 Joules.
To calculate the height the blade would rise if all of the rotational kinetic energy is converted to gravitational potential energy, we can use the conservation of energy:
Rotational energy = Gravitational potential energy
Gravitational potential energy = m * g * h
Where:
- m is the mass of the blade
- g is the acceleration due to gravity (approximately 9.8 m/s²)
- h is the height the blade would rise
Let's calculate the height (h):
2063.21 J = 0.65 kg * 9.8 m/s² * h
Simplifying, we find:
h = 2063.21 J / (0.65 kg * 9.8 m/s²)
= 329.08 m
Therefore, the blade would rise to a height of approximately 329.08 meters.
To find the rotational energy of the blade, we can use the formula:
Rotational Energy = 0.5 * moment of inertia * angular velocity^2
To calculate the moment of inertia of the steel blade, we can use the formula for a rectangular bar rotating about its center:
Moment of Inertia = (1/12) * mass * length^2
Let's start by calculating the moment of inertia:
Mass of the blade (m) = 0.65 kg
Length of the blade (L) = 0.55 m
Moment of Inertia (I) = (1/12) * m * L^2
= (1/12) * (0.65 kg) * (0.55 m)^2
≈ 0.0092 kg·m^2 (rounded to four significant figures)
Next, we need to convert the operating angular speed from rpm to radians per second (rad/s). We know that 1 revolution = 2π radians, and 1 minute = 60 seconds. Therefore:
Angular velocity (ω) = 3450 rpm * (2π rad/1 rev) * (1 rev/60 s)
= 360π rad/s
≈ 1130.97 rad/s (rounded to four significant figures)
Now, we can calculate the rotational energy:
Rotational Energy = 0.5 * I * ω^2
= 0.5 * (0.0092 kg·m^2) * (1130.97 rad/s)^2
≈ 5868 J (rounded to four significant figures)
So, the rotational energy of the blade is approximately 5868 Joules.
To determine the height to which the blade would rise if all the rotational kinetic energy is converted to gravitational potential energy, we can use the following formula:
Gravitational Potential Energy = m * g * h
We know the mass (m) of the blade is 0.65 kg, and the acceleration due to gravity (g) is approximately 9.8 m/s^2. Let's assume the blade starts from rest at ground level (h = 0).
Gravitational Potential Energy = (0.65 kg) * (9.8 m/s^2) * h
= 6.37 J (rounded to two significant figures)
Therefore, if all the rotational kinetic energy of the blade could be converted to gravitational potential energy, it would only rise to a height of approximately 6.37 Joules.