Question

Objects that are at rest relative to Earth's surface are in circular motion due to Earth's rotation. What is the radial acceleration of a painting hanging in the Prado Museum in Madrid, Spain, at a latitude of \(40.2^{\circ}\) North? (Note that the object's radial acceleration is not directed toward the center of the Earth.)

Short answer

The radial acceleration of the painting is approximately 0.0258 m/s².

Step-by-step solution

Understand the Problem

We need to determine the radial (centripetal) acceleration of an object at a given latitude due to Earth's rotation. Here, the object is stationary relative to the Earth but experiences circular motion because of Earth's rotation.

Recall the Formula for Radial Acceleration

The formula for radial acceleration in circular motion is given by \( a_c = \frac{v^2}{r} \), where \( v \) is the tangential velocity and \( r \) is the radius of the circular path. The radial acceleration can also be expressed as \( a_c = r \cdot \omega^2 \), where \( \omega \) is the angular velocity of Earth's rotation.

Determine Earth's Angular Velocity

The angular velocity \( \omega \) of Earth is determined by its rotational period: one full rotation per day (24 hours). Therefore, \( \omega = \frac{2 \pi}{T} \), where \( T = 24 \times 3600 \) seconds is the period. Calculating, \( \omega \approx 7.27 \times 10^{-5} \) radians/second.

Determine Radius of Circular Motion

Given the latitude \( \phi = 40.2^{\circ} \), the effective radius \( r \) for circular motion is given by \( r = R_{\text{earth}} \cdot \cos(\phi) \), where \( R_{\text{earth}} \approx 6371 \) km is Earth's radius. Converting this radius to meters gives about 6,371,000 m.

Calculate the Effective Radius for the Latitude

Calculate: \( r = 6,371,000 \times \cos(40.2^{\circ}) \). Using a calculator, this value becomes approximately 4,872,094 meters.

Compute the Radial Acceleration

Using the formula \( a_c = r \cdot \omega^2 \), substitute the calculated \( r \) and \( \omega \): \( a_c = 4,872,094 \times (7.27 \times 10^{-5})^2 \). Calculate this to find the radial acceleration.

Perform the Final Calculation

Calculating the expression: \( a_c \approx 4,872,094 \times 5.291 \times 10^{-9} \approx 0.0258 \) m/s².

Key concepts

Circular Motion

Objects in circular motion are moving in a path that forms a circle. In this motion, even when the object appears still relative to the rotating frame (like the Earth), it experiences a continuous change in direction. This is what happens to objects due to Earth's rotation.

• The objects maintain a certain distance from the center of rotation, known as the radius of the path.
• In circular motion, this continuous change in direction results in an acceleration directed toward the circle's center — called radial or centripetal acceleration.

Despite the surface of the Earth feeling stationary, every object on it is actually in circular motion because the Earth rotates around its axis. Understanding this helps recognize how radial acceleration affects stationary-seeming objects, like a painting in a museum.

Angular Velocity

Angular velocity refers to how quickly an object rotates or revolves around a center point or axis. It's especially important in calculations involving circular motion, like the motion caused by Earth's rotation.

• In Earth's case, angular velocity is determined by Earth's rotational period — it completes one full spin every 24 hours.
• This is quantified as \[ \omega = \frac{2\pi}{T} \] where \(T\) is the time for one rotation, 86,400 seconds.

Calculating the angular velocity allows us to understand how fast points on Earth, like a museum in Madrid, are "moving" around the axis, even if they seem still. It provides the necessary detail for determining radial acceleration.

Earth's Rotation

Earth's rotation is a fundamental characteristic of our planet, and it affects everything on it. The planet spins around its axis, leading to daily cycles of day and night. This spinning motion also means that any point on Earth, apart from the very poles, is in constant circular motion.

• This rotation causes an outward apparent force that slightly alters gravitational effects experienced on the surface.
• The radial acceleration that results from rotation affects objects depending on their distance from the axis — hence the need to consider latitude.

Understanding Earth's rotation helps explain phenomena like why we have day and night and why there are slight variations in gravity and effective gravitational force experienced at different points on Earth.

Latitude

Latitude measures how far a location is from the equator, running from 0° at the equator to 90° at the poles. This affects the effective radius used in circular motion calculations for Earth.

• The formula for determining the radius, considering latitude, is \[ r = R_{\text{earth}} \cdot \cos(\phi) \] where \(\phi\) is the latitude angle.
• As latitude increases, the effective radius decreases, meaning objects closer to the poles are 'closer' to Earth's axis compared to those near the equator.

At a specific latitude like 40.2°N, the angle provides a means to measure how far the location is from directly spinning on the equator, thus influencing calculations of radial acceleration.