Find the total kinetic energy of a 1500 g wheel, 700 mm in diameter, rolling across a level surface at 650 rpm. Assume that the wheel can be considered as a hoop.
(b) For the wheel in part (a), calculate the torque required to decrease its rotational speed from 400 rpm to rest in 9 s.
To find the total kinetic energy of a rolling wheel, we need to consider both its translational kinetic energy and rotational kinetic energy.
(a) First, let's calculate the translational kinetic energy:
The mass of the wheel is given as 1500 g, which is equivalent to 1.5 kg.
The diameter of the wheel is given as 700 mm, which is equivalent to 0.7 m.
The wheel can be considered as a hoop, which means all its mass is concentrated on its circumference.
The equation for translational kinetic energy is:
Translational Kinetic Energy = (1/2) * mass * velocity^2
To find the velocity of the wheel, we need to convert its rotational speed from rpm (revolutions per minute) to m/s. Since the wheel is rolling without slipping, the linear velocity of a point on its circumference is equal to the product of the angular velocity (in radians per second) and the radius of the wheel.
The equation for converting rpm to rad/s is:
Angular Velocity (in rad/s) = (2π * rotational speed) / 60
Given that the rotational speed is 650 rpm, we can calculate the angular velocity:
Angular Velocity = (2π * 650) / 60 ≈ 68.57 rad/s
The radius of the wheel is half its diameter, so:
Radius = 0.7 m / 2 = 0.35 m
Now, we can calculate the linear velocity:
Linear Velocity = Angular Velocity * Radius = 68.57 rad/s * 0.35 m ≈ 23.9995 m/s (approximately 24 m/s)
Now, substitute the values into the equation for translational kinetic energy:
Translational Kinetic Energy = (1/2) * 1.5 kg * (24 m/s)^2 = 432 J (approximately)
(b) To calculate the torque required to decrease the wheel's rotational speed, we can use the equation:
Torque = Moment of Inertia * Angular Acceleration
Since the wheel can be considered as a hoop, the moment of inertia can be calculated using the formula:
Moment of Inertia = mass * radius^2
Given that the rotational speed needs to decrease from 400 rpm to rest in 9 seconds, the angular acceleration can be calculated as:
Angular Acceleration = (Final Angular Velocity - Initial Angular Velocity) / Time
Converting the rpm values to rad/s:
Initial Angular Velocity = (2π * 400) / 60 ≈ 41.89 rad/s
Final Angular Velocity = 0 rad/s (resting)
Substitute the values into the equation for angular acceleration:
Angular Acceleration = (0 - 41.89 rad/s) / 9 s ≈ -4.65 rad/s^2
Now, we can calculate the moment of inertia:
Moment of Inertia = 1.5 kg * (0.35 m)^2 = 0.3675 kg·m^2
Finally, substitute the values into the equation for torque:
Torque = 0.3675 kg·m^2 * -4.65 rad/s^2 ≈ -1.71 N·m (approximately)