A truck with a total mass of 1.75x103
kg has wheels of radius 32.0 cm and
moment of inertia of 0.800 kg-m2
. Find the truck’s (a) translational energy and (b)
rotational energy when the truck is moving at a speed of 25.0 m/s
To find the truck's translational energy, we can use the formula:
Translational Energy = (1/2) * Mass * Velocity^2
Given:
Mass (m) = 1.75 * 10^3 kg
Velocity (v) = 25.0 m/s
Substituting these values into the formula, we get:
Translational Energy = (1/2) * (1.75 * 10^3 kg) * (25.0 m/s)^2
Translational Energy = (1/2) * (1.75 * 10^3 kg) * (625 m^2/s^2)
Translational Energy = 546875 J
Therefore, the truck's translational energy is 546875 J.
To find the truck's rotational energy, we can use the formula:
Rotational Energy = (1/2) * Moment of Inertia * Angular Velocity^2
Given:
Moment of Inertia (I) = 0.800 kg-m^2
Radius (r) = 32.0 cm = 0.32 m
Velocity (v) = 25.0 m/s
We can find the angular velocity (ω) using the formula:
Velocity (v) = Radius (r) * Angular Velocity (ω)
Substituting the known values:
25.0 m/s = 0.32 m * ω
ω = 25.0 m/s / 0.32 m
ω = 78.125 rad/s
Now, substituting the values into the formula for rotational energy:
Rotational Energy = (1/2) * (0.800 kg-m^2) * (78.125 rad/s)^2
Rotational Energy = (1/2) * (0.800 kg-m^2) * (6105.47 rad^2/s^2)
Rotational Energy = 2442.19 J
Therefore, the truck's rotational energy is 2442.19 J.
To find the truck's translational energy and rotational energy, we need to use the formulas for each type of energy.
(a) Translational Energy:
The translational energy of an object can be calculated using the formula:
Translational Energy (ET) = 1/2 * m * v^2
where m is the mass of the truck and v is the speed of the truck.
Given:
Mass of the truck (m) = 1.75 x 10^3 kg
Speed of the truck (v) = 25.0 m/s
Plugging in the values into the formula, we have:
ET = 1/2 * (1.75 x 10^3 kg) * (25.0 m/s)^2
Calculating this, we get:
ET = 1/2 * (1.75 x 10^3 kg) * (625.0 m^2/s^2)
ET = 546,875 J
Therefore, the truck's translational energy is 546,875 Joules.
(b) Rotational Energy:
The rotational energy of an object can be calculated using the formula:
Rotational Energy (ER) = 1/2 * I * ω^2
where I is the moment of inertia of the truck's wheels and ω is the angular velocity of the wheels.
Given:
Moment of inertia of the wheels (I) = 0.800 kg-m^2
Radius of the wheels (r) = 32.0 cm = 0.32 m
Speed of the truck (v) = 25.0 m/s
To find the angular velocity (ω), we can use the relation between linear velocity (v) and angular velocity (ω):
v = ω * r
Rearranging the equation, we have:
ω = v / r
ω = (25.0 m/s) / (0.32 m)
ω = 78.125 rad/s
Plugging in the values into the formula for rotational energy, we have:
ER = 1/2 * (0.800 kg-m^2) * (78.125 rad/s)^2
Calculating this, we get:
ER = 1/2 * (0.800 kg-m^2) * (6,109.375 rad^2/s^2)
ER = 2,444.75 J
Therefore, the truck's rotational energy is 2,444.75 Joules.
v = 25 m/s
so
25 = omega R = omega * 0.32
omega = 25 / 0.32 radians/second
I = 4 * 0.8 = 3.2 kg m^2
Translational = (1/2) m v^2
Rotational = (1/2) I omega^2