Problem 31 - College Physics

Question

\(\bullet\) You swing a 2.2 \(\mathrm{kg}\) stone in a circle of radius 75 \(\mathrm{cm} .\) At what speed should you swing it so its centripetal acceleration will be 9.8 \(\mathrm{m} / \mathrm{s}^{2}\) ?

Step-by-Step Solution

Answer

The speed should be approximately 2.71 m/s.

Step 1: Conceptualize Centripetal Acceleration

Centripetal acceleration (\( a_c \)) is the acceleration that keeps an object moving in a circular path. It always points towards the center of the circle, and its formula is given by \( a_c = \frac{v^2}{r} \), where \( v \) is the speed of the object and \( r \) is the radius of the circle.

Step 2: Identify Given Values

In the problem, the given values are the mass of the stone \( m = 2.2 \, \mathrm{kg} \), the radius \( r = 75 \, \mathrm{cm} \), and the centripetal acceleration \( a_c = 9.8 \, \mathrm{m/s^2} \). Note that we need to convert the radius to meters by dividing by 100: \( r = 0.75 \, \mathrm{m} \).

Step 3: Rearrange the Formula for Speed

We need to find the speed \( v \). The formula for centripetal acceleration is \( a_c = \frac{v^2}{r} \). Solving for \( v \), we get \( v = \sqrt{a_c \times r} \).

Step 4: Substitute Values into the Formula

Substitute \( a_c = 9.8 \, \mathrm{m/s^2} \) and \( r = 0.75 \, \mathrm{m} \) into the rearranged formula:\[ v = \sqrt{9.8 \times 0.75} \].

Step 5: Calculate the Speed

Perform the calculation:\[ v = \sqrt{7.35} \approx 2.71 \, \mathrm{m/s} \].

Key Concepts

Circular Motion

Circular motion refers to the movement of an object along a circular path. This kind of motion is defined by its curvature, meaning the path is curved rather than straight. To maintain this circular path, a force known as centripetal force is required. This force is always directed towards the center of the circle. In the given exercise, the stone moving in a circle is an example of circular motion.

Understanding circular motion involves a few key ideas:

• Centripetal Force: This is the force that keeps an object moving in a circle. Without it, the object would fly off in a straight line due to inertia. It's always directed towards the center of the circle.
• Centripetal Acceleration: As we analyzed in the original problem, this is the acceleration that occurs because of the change in direction of the velocity vector. It's expressed through the formula: \( a_c = \frac{v^2}{r} \), where \( v \) is the object's speed and \( r \) is the radius of the circle.

The movement of the stone in our problem is dictated by these forces. The centripetal acceleration we calculated signifies how quickly the stone needs to change direction at any point in its path.

Speed Calculation

In the context of circular motion, speed is essential for determining how an object navigates along its circulatory path. The problem specifically involves calculating the speed needed to achieve a specific centripetal acceleration, which in this case is 9.8 m/s².

The formula we use is derived from the definition of centripetal acceleration: \( a_c = \frac{v^2}{r} \). Solving for speed, we rearrange to get \( v = \sqrt{a_c \times r} \). This equation tells us that the speed is dependent on both the desired acceleration and the radius of the circle.

• Radius Conversion: In problems, the radius must always be in meters for consistency in units. For our exercise, the radius was initially given as 75 cm, which we converted to 0.75 m.
• Substitution and Calculation: We then substituted the given values into our rearranged formula. The speed \( v \) was calculated as \( \sqrt{9.8 \times 0.75} \), resulting in approximately 2.71 m/s.

Proper calculation of speed is crucial because it ensures that the object has the right amount of momentum to stay on its circular track without deviating.

Physics Problem Solving

When solving physics problems, especially involving circular motion, following a systematic approach can make things much easier.

Let's break down the steps used in the solution, which can be applied to similar challenges:

• Conceptualization: Start by understanding the physical concepts involved. For circular motion, grasp what causes centripetal force and acceleration.
• Identify Known Variables: List all given values, like mass, radius, and centripetal acceleration, and ensure they're in compatible units.
• Formula Rearrangement: Use known formulas like \( a_c = \frac{v^2}{r} \) and rearrange them to isolate the desired variable, such as speed in our problem.
• Substitution and Computation: Substitute the known values into the formula and perform the arithmetic carefully to get the solution.

By following these steps, physics problems become less daunting, allowing for a clear pathway from understanding the problem to finding a solution. Precision in calculation and clarity in the application of formulas are key to success.