Question

A ball attached to the end of a string is swung around in a circular path of radius $r$. If the radius is doubled and the linear speed is kept constant, the centripetal acceleration a) remains the same. b) increases by a factor of 2 . c) decreases by a factor of 2 . d) increases by a factor of 4 e) decreases by a factor of 4 .

Short answer

Question: When a ball swings around in a circular path with a constant linear speed and the radius is doubled, the centripetal acceleration: a) stays the same b) increases by a factor of 2 c) decreases by a factor of 2 d) increases by a factor of 4 Answer: c) decreases by a factor of 2.

Step-by-step solution

Recall the formula for centripetal acceleration

The formula for centripetal acceleration is given by: $a_c=\frac{v^2}{r}$ where $a_c$ is the centripetal acceleration, $v$ is the linear speed, and $r$ is the radius of the circular path.

Establish the initial condition

Initially, the ball is swinging with certain centripetal acceleration, $a_{c1}$, linear speed $v$, and radius $r$: $a_{c1}=\frac{v^2}{r}$

Examine the change in radius

Now the radius is doubled. Let the new radius be $2r$.

Determine the new centripetal acceleration

Since the linear speed remains constant, we can find the new centripetal acceleration, $a_{c2}$, using the doubled radius $2r$: $a_{c2}=\frac{v^2}{2r}$

Compare the initial and new centripetal acceleration

Now that we have the expressions for both the initial and new centripetal acceleration, we can compare them: From step 2, we have $a_{c1}=\frac{v^2}{r}$ And from step 4, we have $a_{c2}=\frac{v^2}{2r}$ To find the factor by which the centripetal acceleration has changed, we can divide $a_{c2}$ by $a_{c1}$: $\frac{a_{c2}}{a_{c1}}=\frac{\frac{v^2}{2r}}{\frac{v^2}{r}}=\frac{1}{2}$ This shows that the centripetal acceleration has decreased by a factor of 2. So the correct answer is: c) decreases by a factor of 2.

Key concepts

Circular Motion

Circular motion is a fascinating concept in physics that refers to the movement of an object along the circumference of a circle. This kind of motion is quite common in our day-to-day lives, like when a car goes around a circular track or a planet orbits the sun. When an object moves in a circular path, it constantly changes its direction, which implies that there's an acceleration involved, known as centripetal acceleration. This acceleration is directed towards the center of the circle, ensuring the object's path remains circular.

Some important features of circular motion include:

• **Constant speed**: In uniform circular motion, the object travels with a steady speed, but because its direction changes constantly, it experiences acceleration.
• **Centripetal force**: This is the force responsible for keeping an object moving in a circle. It's always directed towards the center of the circle.
• **Radius**: The distance from the center of the circle to the path of the object. It plays a crucial role in determining the centripetal force and acceleration.

Understanding these components helps in solving problems related to objects moving in circular paths efficiently.

Physics Problem Solving

Solving physics problems, especially those involving circular motion, requires a systematic approach. At times, students find it challenging to connect the theoretical framework to practical scenarios. Here's a simple method to solve such problems effectively:

• **Understand the problem**: Read the problem carefully and identify what is given and what needs to be found.
• **Use relevant formulas**: In problems involving circular motion, key formulas like the centripetal acceleration formula, $a_c=\frac{v^2}{r}$, are essential.
• **Break down the steps**: Solve the problem step-by-step, like determining the initial conditions first before applying any changes, such as a new radius or speed.
• **Check your work**: After solving, verify if the solution makes sense logically. For instance, if a radius is doubled, comprehending how this impacts acceleration will confirm your solution.

A thoughtful and organized method significantly enhances accuracy when tackling complex physics challenges.

Linear Speed and Radius Relationship

In circular motion, the relationship between linear speed and radius is crucial when considering centripetal acceleration. Let's delve into how changes in these factors alter the circular motion dynamics.

Keeping linear speed constant, as seen in the given exercise, let's explore the effect of a changing radius:

• **Impact of increased radius**: If the radius is increased while keeping the linear speed fixed, the centripetal acceleration decreases since the denominator of the formula $a_c=\frac{v^2}{r}$ gets larger.
• **Result on centripetal acceleration**: Doubling the radius results in the centripetal acceleration cutting in half. Hence, there's an inverse relationship between the radius and centripetal acceleration, assuming speed remains steady.
• **Practical implications**: Understanding this relationship helps in designing systems where circular motion is involved, like in car or rollercoaster designs, ensuring safety and functionality.

Recognizing how linear speed and radius interact allows for better control over objects in circular motion, which is vital in both theoretical applications and real-world physics scenarios.