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Chapter 4: Problem 34

An automobile whose speed is increasing at a rate of 0.600 \(\mathrm{m} / \mathrm{s}^{2}\) travels along a circular road of radius 20.0 \(\mathrm{m}\) . When the instantaneous speed of the automobile is 4.00 \(\mathrm{m} / \mathrm{s}\) , find (a) the tangential acceleration component, (b) the centripetal acceleration component, and \((\mathrm{c})\) the magnitude and direction of the total acceleration.

Short Answer

Expert verified
The tangential acceleration is 0.600 m/s^2, the centripetal acceleration is 0.800 m/s^2, and the magnitude of the total acceleration is approximately 1.00 m/s^2 directed at an angle of 53.1 degrees relative to the tangential acceleration.

Step by step solution

01

Identify the Tangential Acceleration Component

The tangential acceleration is given directly by the rate of increase in speed, which is 0.600 m/s^2.
02

Calculate the Centripetal Acceleration Component

Use the formula for centripetal acceleration, which is ac = v^2 / r, where ac is the centripetal acceleration, v is the instantaneous speed, and r is the radius of the circular path.
03

Compute the Centripetal Acceleration

Substitute the given values into the centripetal acceleration formula to find ac: ac = (4.00 m/s)^2 / 20.0 m.
04

Determine the Magnitude of the Total Acceleration

Use the Pythagorean theorem since the tangential and centripetal accelerations are perpendicular to each other: total acceleration (a) = sqrt(at^2 + ac^2), where at is tangential acceleration and ac is centripetal acceleration.
05

Find the Direction of the Total Acceleration

The direction of the total acceleration is given by the angle theta relative to the tangential acceleration, where tan(theta) = ac/at. Thus, theta = arctan(ac/at).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tangential Acceleration
Imagine you're in a car that's speeding up as it moves around a curve. The feeling of being pushed back into your seat results from tangential acceleration. It's the rate at which the speed of an object changes as it travels along a straight path or a curve. In our scenario with the automobile, the tangential acceleration is constant at 0.600 m/s2, indicating a steady increase in speed.

Tangential acceleration is always directed along the tangent to the path of the object. This means it points in the direction of the velocity's change. So, if you think about the automobile in our exercise, the tangential acceleration is directly increasing the car's speed while maintaining its direction along the road.
Centripetal Acceleration
Now, let's focus on the sensation you experience while turning sharply in a vehicle, like you're being pulled toward the side. That is centripetal acceleration at work. It's the inward acceleration necessary for an object to move in a circular path and is directed towards the center of the circle. You can calculate centripetal acceleration with the formula \( a_c = \frac{v^2}{r} \), where \( v \) is the instantaneous speed, and \( r \) is the radius of the circular path.

In our problem, the automobile traveling at 4.00 m/s along the 20.0 m radius curve experiences centripetal acceleration. It's important to understand that centripetal acceleration does not change the object's speed but alters its direction, keeping it moving in a curve rather than a straight line.
Total Acceleration
When simultaneous accelerations occur 鈥� like the tangential acceleration from speeding up and the centripetal acceleration from the curve 鈥� we find the total acceleration. It's the vector sum of all acceleration components involved in the motion. Since tangential and centripetal accelerations are perpendicular to each other, we use the Pythagorean theorem to determine the total acceleration: \( a = \sqrt{a_t^2 + a_c^2} \).

The total acceleration combines the car's increased speed and change in direction into a single vector. If you could feel this as a passenger, it's like being pressed back into your seat while simultaneously being tugged sideways into your seatbelt.
Instantaneous Speed
Instantaneous speed is the speed of an object at a specific moment in time. Unlike average speed, which tells you how fast something is going over a period of time, instantaneous speed is all about 'the now.' It's like glancing at your car's speedometer to see exactly how fast you're traveling at that second.

In the exercise context, the instantaneous speed of 4.00 m/s is what the car's speedometer would show at the precise moment we're analyzing. This value becomes crucial for calculating centripetal acceleration and, together with tangential acceleration, helps to find the total acceleration of the car.

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