Question

A small spherical steel ball is placed at a distance slightly away from the center of a concave mirror having radius of curvature \(250 \mathrm{~cm}\). If the ball is released, it will now move on the curved surface. What will be the periodic time of this motion? Ignore frictional force and take \(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\). (A) \((\pi / 4) \mathrm{s}\) (B) \(\pi \mathrm{s}\) (C) \((\pi / 2) \mathrm{s}\) (D) \(2 \pi \mathrm{s}\)

Short answer

The short answer is: The periodic time of the motion is \(\pi \mathrm{s}\) (Answer choice B).

Step-by-step solution

Determine the length of the pendulum

The length of the pendulum is equivalent to the radius of curvature of the concave mirror, which is given as \(250 \mathrm{~cm}\). Since we need to use the value of \(\mathrm{g}\) in \(\mathrm{m/s}^2\), we will convert the length \(L\) into meters. So, \(L = 2.5 \mathrm{~m}\).

Use the formula for finding the periodic time of a simple pendulum

For a simple pendulum, the periodic time \(T\) can be given by the formula: \[T = 2\pi\sqrt{\frac{L}{g}}\] where \(L\) is the length (radius of curvature) in meters and \(g\) is the acceleration due to gravity in \(\mathrm{m/s}^2\).

Calculate the periodic time

Plug in the given values and calculate the periodic time: \[T = 2\pi\sqrt{\frac{2.5}{10}}\] \[T = 2\pi\sqrt{\frac{1}{4}}\] \[T = 2\pi\left(\frac{1}{2}\right)\] \[T = \pi \mathrm{s}\] The periodic time of the motion is \(\pi \mathrm{s}\), which corresponds to the answer choice (B).

Key concepts

Concave Mirror

A concave mirror is a type of spherical mirror with an inwardly curved reflecting surface. Unlike flat mirrors, concave mirrors can focus light inward toward a single point, known as the focal point. This makes them unique and capable of producing a variety of image types. In essence, they can either magnify or diminish an object depending on the position of the object relative to the mirror.

The radius of curvature is the distance from the mirror's surface to its center of curvature. For a concave mirror, this radius determines the focal length, specifically, the focal point is located halfway to the center of curvature. For example, if a concave mirror has a radius of curvature of 250 cm, the focal length is half of this, meaning 125 cm. This is important because the position of images formed by the mirror depends greatly on these measurements.

Concave mirrors also have a wide range of practical applications, from telescopes and shaving mirrors to headlamps. In physics problems, they are often used to explore concepts of reflection and refraction through their ability to manipulate light paths.

Periodic Time

Periodic time refers to the time required for one complete cycle of oscillation to occur. In the context of a simple pendulum, it is the time it takes to swing from one side, to the furthest point on the opposite side, and back again. This time, known as the "period" (T), is a crucial aspect of understanding harmonic motion.

For a simple pendulum, the period can be calculated using the formula:\[T = 2\pi\sqrt{\frac{L}{g}}\]
Here:

• L is the length of the pendulum (or distance from the pivot point to the center of mass)
• g is the acceleration due to gravity

Notably, the period is independent of both the mass of the pendulum and the amplitude (assuming small angle oscillations). This means that for a given length and gravitational field, the period remains constant.

The problem in question uses this formula to determine that the periodic time is \(\pi\, \text{seconds}\). This calculation assumes there is no friction or air resistance affecting the motion.

Radius of Curvature

In geometric terms, the radius of curvature is the distance from the center of curvature to a point on the curve or surface. In physics, especially with mirrors and lenses, it is a critical property that affects how light and images are projected. For a concave mirror, this distance determines how an object will appear and, in the case of a simple pendulum, it equates to the length of the pendulum.

When calculating for the periodic time of a pendulum using a concave mirror, the radius of curvature transforms into the pendulum's effective length. As shown in the exercise, a radius of curvature of 250 cm translates to a pendulum length of 2.5 m after conversion to meters. This length then plugs into the formula for the pendulum’s periodic time, illustrating how physical dimensions directly influence oscillatory behavior.

Acceleration Due to Gravity

Acceleration due to gravity (\(g\)) is a fundamental concept in physics, representing the force with which the Earth pulls objects toward its center. This constant is approximately 9.81 m/s² on the surface of the Earth but is often rounded to 10 m/s² for simplicity in calculations.

In pendulum motion, \(g\) plays a critical role in determining the period of oscillation. Since \(T = 2\pi\sqrt{\frac{L}{g}}\), the value of \(g\) inversely affects the period. Thus, greater gravity results in a shorter period, while weaker gravity lengthens the period.

This dependency highlights why gravity is essential in pendulum timing experiments. When solving physics problems like the example provided, assuming a standard \(g\) value of 10 m/s² enables straightforward and accurate computations.