A child's playground ride consists of four seats, of mass 12kg each, connected to a vertical axle with spokes of small mass. The seats are placed equidistant in a circle of radius... A child's playground ride consists of four seats, of mass 12kg each, connected to a vertical axle with spokes of small mass. The seats are placed equidistant in a circle of radius 1.8m and rotate about the vertical axle. A child of mass 21kg sits in one of the seats and his friend pushes the ride to accelerate him from rest to 0.6 rev/s. How much work does the friend do?
Understand the Problem
The question is asking us to calculate the work done by a friend who is pushing a child's playground ride to accelerate it to a specific rotational speed. To solve this, we need to find the moment of inertia of the system and then use the work-energy principle to find the work done during the acceleration.
Answer
The work done by the friend is calculated using the equation $W = \frac{1}{2} I \omega^2$.
Answer for screen readers
The work done, $W$, is given by:
$$ W = \frac{1}{2} I \omega^2 $$
Substituting the previously calculated $I$ and $\omega$ will give the final value.
Steps to Solve
- Calculate the Moment of Inertia (I)
To find the moment of inertia for the child's playground ride, we need its mass and the radius of rotation. The moment of inertia for a point mass is given by the formula:
$$ I = m \cdot r^2 $$
where $m$ is the mass and $r$ is the radius of rotation.
- Determine the Angular Velocity (ω)
Next, we need to determine the final angular velocity we want to achieve. The angular velocity can be calculated if we know the number of revolutions per minute (RPM) and then convert it to radians per second:
$$ \omega = \frac{2\pi \text{ RPM}}{60} $$
- Apply the Work-Energy Principle
The work-energy principle states that the work done on an object is equal to the change in kinetic energy. The rotational kinetic energy (KE) of a body is given by:
$$ KE = \frac{1}{2} I \omega^2 $$
So the work done (W) can be expressed as:
$$ W = KE = \frac{1}{2} I \omega^2 $$
Substituting the moment of inertia calculated earlier into this equation gives us the work done.
- Calculate the Work Done
Finally, plug in the values for moment of inertia and angular velocity into the work equation to calculate the work done by the friend in accelerating the ride.
The work done, $W$, is given by:
$$ W = \frac{1}{2} I \omega^2 $$
Substituting the previously calculated $I$ and $\omega$ will give the final value.
More Information
To find the work done to accelerate an object, we need to consider both its rotational inertia and the speed at which it is rotated. This principle is fundamental in mechanics, especially when dealing with rotational dynamics.
Tips
- Forgetting to convert RPM to radians per second.
- Using incorrect formulas for moment of inertia depending on the shape of the object.
- Not squaring the angular velocity when calculating kinetic energy.
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