Problem 113 - University Physics with Modern Physics

Question

One December identical twins Jena and Jackie are playing on a large merry-go- round (a disk mounted parallel to the ground, on a vertical axle through its center) in their school playground in northern Minnesota. Each twin has mass 30.0 \(\mathrm{kg}\) . The icy coating on the merry-go-round surface makes it frictionless. The merry-go-round revolves at a constant rate as the twins ride on it. Jena, sitting 1.80 \(\mathrm{m}\) from the center of the merry-go-round, must hold on to one of the metal posts attached to the merry-go-round with a horizontal force of 60.0 \(\mathrm{N}\) to keep from sliding off. Jackie is sitting at the edge, 3.60 \(\mathrm{m}\) from the center. (a) With what horizontal force must Jackie hold on to keep from falling off? (b) If Jackie falls off, what will be her horizontal velocity when she becomes airborne?

Step-by-Step Solution

Answer

(a) 120 N; (b) 7.2 m/s

Step 1: Calculate Centripetal Force for Jena

To find the centripetal force exerted on Jena, use the formula:\[F_c = m \cdot a_c\]where \(F_c\) is the centripetal force, \(m\) is the mass, and \(a_c\) is the centripetal acceleration. Jena needs 60 N force to stay on the merry-go-round. Therefore, the centripetal acceleration \(a_c\) can be calculated using:\[a_c = \frac{F_c}{m} = \frac{60 \text{ N}}{30.0 \text{ kg}} = 2 \text{ m/s}^2\]

Step 2: Calculate Angular Velocity

Since centripetal acceleration \(a_c\) also equals \(r \cdot \omega^2\) where \(r\) is the radius (distance from center) and \(\omega\) is the angular velocity, we use Jena's position:\[a_c = r \cdot \omega^2\]Substituting for \(a_c\) and \(r\) for Jena:\[2 = 1.80 \cdot \omega^2 \Rightarrow \omega^2 = \frac{2}{1.80} \Rightarrow \omega = \sqrt{\frac{2}{1.80}}\]

Step 3: Calculate Centripetal Force for Jackie

Jackie is sitting at 3.60 m from the center. Using \(a_c = r \cdot \omega^2\), and knowing \(\omega\) from Jena's calculation:\[a_c = 3.60 \times \frac{2}{1.80} = 4 \text{ m/s}^2\]The force required for Jackie:\[F_{c, Jackie} = m \cdot a_c = 30.0 \times 4 = 120 \text{ N}\]

Step 4: Calculate Horizontal Velocity for Jackie when Airborne

When Jackie falls off, she continues moving with the linear velocity given by \(v = \omega \cdot r\). From previous steps, \(\omega^2 = \frac{2}{1.80}\), hence:\[v = \sqrt{\frac{2}{1.80}} \times 3.60\]Simplifying gives:\[v = \sqrt{\frac{4}{1.80}} \times 3.60 = 2.0 \times 3.60 = 7.2 \text{ m/s}\]

Key Concepts

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around a central point or axis. It describes the rate of rotation and is usually represented by the Greek letter \( \omega \). Angular velocity is different from linear velocity because it specifically measures rotational motion.
In the problem, we use the concept of angular velocity to determine how the merry-go-round spins as the twins hold on. For Jena, sitting at a radius of 1.80 meters, her centripetal acceleration \( a_c \) is linked to the angular velocity using the formula:

• \( a_c = r \cdot \omega^2 \)

We calculated Jena’s centripetal acceleration as 2 m/s², and inserted it into the formula to find \( \omega \). Thus, \( \omega = \sqrt{\frac{2}{1.80}} \).
This relationship helps us determine the spinning rate essential to find the forces acting on both Jena and Jackie as they cling to the merry-go-round.

Centripetal Acceleration

Centripetal acceleration is the rate at which an object's velocity changes as it moves along a circular path. Even at a constant speed, as in our merry-go-round problem, a changing direction equals acceleration. This type of acceleration is always directed towards the center of the circle in which the object is moving.
For both Jena and Jackie, knowing the centripetal acceleration is critical to assess the force required to keep them on the merry-go-round. Jena's centripetal acceleration is calculated using the force she exerts to hold on:

• \( a_c = \frac{F_c}{m} \)

Where \( F_c = 60 \text{ N} \) and \( m = 30.0 \text{ kg} \), resulting in \( a_c = 2 \text{ m/s}^2 \).
With Jena's centripetal acceleration known, we are able to calculate the angular velocity which is crucial for solving Jackie's problem.

Physics Problem Solving

Physics problems, like the merry-go-round scenario, often require applying concepts systematically to find the desired results. In our case, we used step-by-step reasoning to solve for forces and velocities. We began with what we know and applied relevant formulas to solve for what we didn’t know.
Effective problem-solving in physics involves:

• Understanding the situation and visualizing the physical setup.
• Identifying known and unknown variables from the problem statement.
• Using appropriate physics equations and formulas to link the knowns to the unknowns.
• Applying logical mathematical steps to derive solutions.

In our exercise, identifying the forces and acceleration helped calculate the centripetal force Jackie needs to resist and the velocity she would have if she became airborne.