Problem 134 - Fundamentals of Physics

Question

A car travels around a flat circle on the ground, at a constant speed of \(12.0 \mathrm{~m} / \mathrm{s}\). At a certain instant the car has an acceleration of \(3.00 \mathrm{~m} / \mathrm{s}^{2}\) toward the east. What are its distance and direction from the center of the circle at that instant if it is traveling (a) clockwise around the circle and (b) counterclockwise around the circle?

Step-by-Step Solution

Answer

(a) 48 m west of center; (b) 48 m east of center.

Step 1: Understanding Circular Motion

When an object moves in a circular path at constant speed, it has a centripetal acceleration directed towards the circle's center. This acceleration can be calculated using the formula \( a_c = \frac{v^2}{r} \), where \( v \) is the speed and \( r \) is the radius of the circle.

Step 2: Determine Centripetal Acceleration

Given the speed of the car as \( v = 12.0 \ \mathrm{m/s} \), and the centripetal acceleration \( a_c = 3.00 \ \mathrm{m/s}^2 \), we use the formula \( a_c = \frac{v^2}{r} \) to solve for \( r \). Substituting the known values: \( 3.00 = \frac{12^2}{r} \).

Step 3: Solve for the Radius

Solving for \( r \), we rearrange the equation: \( r = \frac{12^2}{3.00} = \frac{144}{3.00} = 48 \ \mathrm{m} \). This gives the radius of the circle.

Step 4: Determine Direction of Acceleration

Given that the car's instantaneous acceleration toward the east at \( 3.00 \mathrm{~m/s}^2 \), this indicates the direction of the centripetal force, which is inward to the circle's center.

Step 5: Direction of the Car's Motion

(a) If the car is moving clockwise, the inward acceleration toward the center means the car is located west of the circle's center at that instant. (b) If moving counterclockwise, the car is located east of the circle's center.

Key Concepts

Centripetal Acceleration

Centripetal acceleration is a fundamental part of motion in a circle. It acts towards the center of the circle, keeping an object moving along a circular path. Even though the object travels at a constant speed, it continually changes direction, necessitating a net force towards the center. This force is known as the centripetal force.

The formula to calculate centripetal acceleration is given by \[ a_c = \frac{v^2}{r} \]where:

• \( a_c \) is the centripetal acceleration.
• \( v \) is the speed of the object.
• \( r \) is the radius of the circle.

Understanding centripetal acceleration helps explain why an object doesn't just fly off tangentially. It is constantly being "pulled" inward by the centripetal force, maintaining its circular path.

Constant Speed

When an object moves with constant speed along a circular path, its velocity remains unchanged in magnitude. However, it's essential to note that the velocity's direction continually changes to enable circular motion.

Thus, even though the speed is constant, the object experiences an acceleration, because acceleration depends not only on changes in speed but also on changes in direction. This is why centripetal acceleration is a key concept in circular motion. Maintaining constant speed in circular motion involves continuous energy input to counteract any frictional forces, ensuring the speed remains steady while direction changes.

Direction of Motion

In circular motion, direction is constantly changing, which is what defines the movement as circular. The direction of motion is tangential to the circle at any point, meaning it is perpendicular to the radius. In understanding the car's motion, take note:

• For clockwise motion, if the centripetal acceleration is directed east, the car is located west relative to the circle's center at that moment.
• For counter-clockwise motion, the car will be positioned east if the centripetal acceleration points east.

This understanding helps in determining the car's position in relation to the circle's center, based on which way it is moving at that point in time.

Radius of Circle

The radius of the circle is a vital aspect of circular motion. It represents the distance from the center of the circle to any point on the circle's edge.The radius is deeply tied to both centripetal acceleration and the speed of the moving object. As seen in the formula \[ r = \frac{v^2}{a_c} \]where:

• \( r \) is the radius.
• \( v \) is the velocity or speed.
• \( a_c \) is the centripetal acceleration.

In our problem, with a constant speed of \( 12.0 \, \text{m/s} \) and an acceleration of \( 3.00 \, \text{m/s}^2 \), the radius calculates to \( 48 \, \text{m} \). A larger radius means a broader circle, affecting how an object navigates its path.