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. d) increases by a factor of 4 . b) increases by a factor of 2 . e) decreases by a factor of 4 c) decreases by a factor of 2 .

Short answer

Answer: The centripetal acceleration decreases by a factor of 2.

Step-by-step solution

Identifying the centripetal acceleration formula

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

Setting up the initial situation

We are given that the initial radius is r and the linear speed v is kept constant. The initial centripetal acceleration can be calculated using the formula: \[a_{c1} = \frac{v^2}{r}\]

Setting up the modified situation

Now, we will double the radius and keep the linear speed constant. So, the new radius is \(2r\). The new centripetal acceleration can be calculated using the formula: \[a_{c2} = \frac{v^2}{2r}\]

Comparing the initial and final centripetal accelerations

Now, let's find the ratio of the final centripetal acceleration to the initial centripetal acceleration: \[\frac{a_{c2}}{a_{c1}} = \frac{v^2/(2r)}{v^2/r} = \frac{1}{2}\]

Concluding the result

Since the ratio is 1/2, we can conclude that the centripetal acceleration when the radius is doubled and the linear speed is kept constant decreases by a factor of 2. The correct answer is choice (c).

Key concepts

Uniform Circular Motion

When an object moves in a circular path at a constant speed, it's said to be in uniform circular motion. Despite the speed being constant, the object is constantly changing direction, which means it is accelerating. This acceleration is always directed towards the center of the circle, hence it's called centripetal acceleration. The presence of this acceleration indicates that a force must be acting towards the center to keep the object in its circular path, known as the centripetal force. This is similar to how a ball tied to a string is swung around; the tension in the string provides the necessary centripetal force for uniform circular motion.

Interestingly, the magnitude of the centripetal acceleration is determined by the object's linear speed and the radius of the path it follows. If either of these variables is altered while the other is kept constant, the centripetal acceleration will change accordingly. In our example exercise, the ball maintains the same linear speed even as the radius of its path is doubled.

Linear Speed

The term linear speed refers to the rate at which an object covers distance. It's a scalar quantity, meaning it only has magnitude and not a direction. In the context of circular motion, linear speed is the speed at which the object moves along the circular path.

Importance of Linear Speed in Centripetal Acceleration

As shown by the formula for centripetal acceleration \(a_c = \frac{v^2}{r}\), linear speed \(v\) is squared. This means that any change in the linear speed of an object in uniform circular motion will have a significant impact on its centripetal acceleration. If linear speed increases, the centripetal acceleration increases by the square of the factor by which the speed increased. Conversely, if the speed decreases, the centripetal acceleration decreases accordingly.

However, in our exercise, linear speed remains constant when the radius of the circular path changes. Thus, the only variable affecting the centripetal acceleration is the radius, which makes this particular exercise a great demonstration of how centripetal acceleration depends inversely on the radius when speed is constant.

Radius of Circular Path

The radius of the circular path is the distance from the center of the circle to any point on its circumference. In uniform circular motion, this distance remains constant, which defines the size of the path that the object follows.

Role of Radius in Centripetal Acceleration

The formula \(a_c = \frac{v^2}{r}\) indicates that centripetal acceleration is inversely proportional to the radius of the circular path. When the radius is doubled, as in our textbook problem, the centripetal acceleration is halved, given that the linear speed does not change. This inverse relationship means that for a larger circular path, less acceleration is required to maintain the circular motion at the same speed.

In our exercise, a calculation demonstrates this principle: as the radius \(r\) is increased to \(2r\), the centripetal acceleration decreases by a factor of 2. This scenario illustrates a fundamental aspect of circular motion - the larger the radius, the smaller the acceleration needed to keep the object moving in its path, assuming other factors like linear speed remain constant.