a bolt located 50mm from the center of an auto wheel is tightened by the couple shown for .10 seconds. determine the resulting angular velocity of the wheel if the wheel has a radius of gyration of 250 mm and a mass of 19 kilograms
To determine the resulting angular velocity of the wheel, we can use the principle of conservation of angular momentum.
Angular momentum is given by the equation: Angular momentum = moment of inertia * angular velocity.
The given information provides us with the distance of the bolt from the center of the wheel (50mm), the radius of gyration of the wheel (250mm), and the mass of the wheel (19kg).
First, we need to find the moment of inertia of the wheel. The moment of inertia of a rigid body is a measure of its resistance to rotational motion and depends on its mass distribution. In this case, we can use the formula for the moment of inertia of a solid disk:
Moment of inertia (I) = (1/2) * m * r^2,
where m is the mass of the wheel and r is the radius of gyration.
Plugging in the values, we get:
I = (1/2) * 19 kg * (0.250 m)^2
= 0.594 kg * m^2
Now, let's find the torque applied to the wheel. Torque is a measure of the turning force around an axis and is calculated as the product of force and the moment arm (distance from the axis of rotation).
Given that the couple applied to the wheel results in a torque, and the torque is defined as the product of force (F) and the distance from the center of rotation (r), the torque is:
Torque (τ) = force (F) * distance (r)
In this case, the distance is given as 50mm = 0.050m.
The torque is given for a duration of 0.10 seconds, so we can determine the average force applied using the equation:
τ = F * r,
F = τ / r.
Plugging in the values, we have:
F = τ / r
= 0.050 Nm / 0.050 m
= 1 N.
Now, we can use the principle of conservation of angular momentum to find the resulting angular velocity.
The initial angular momentum of the wheel is zero since it is not rotating initially. Therefore, the change in angular momentum is equal to the torque applied multiplied by the duration:
Change in angular momentum = torque * duration
= τ * Δt.
Substituting in the values:
Change in angular momentum = (1 N)(0.10 s)
= 0.10 Nm·s.
Since angular momentum is the product of the moment of inertia (I) and the final angular velocity (ω), we have:
Change in angular momentum = I * ω.
Plugging in the values, we can now solve for ω:
0.10 Nm·s = (0.594 kg·m^2) * ω.
Rearranging the equation, we get:
ω = (0.10 Nm·s) / (0.594 kg·m^2)
≈ 0.168 rad/s.
Therefore, the resulting angular velocity of the wheel is approximately 0.168 radians per second.