Problem 85 - Physics for Scientists and Engineers with Modern Physics

Question

Determine the tangential and centripetal components of the net force exerted on a car (by the ground) when its speed is \(27 \mathrm{~m} / \mathrm{s}\), and it has accelerated to this speed from rest in \(9.0 \mathrm{~s}\) on a curve of radius \(450 \mathrm{~m}\). The car's mass is \(1150 \mathrm{~kg}\).

Step-by-Step Solution

Answer

The tangential force is 3450 N, and the centripetal force is 1863 N.

Step 1: Determine the Tangential Acceleration

The tangential acceleration (\(a_t\)) is the change in speed over time. Given that the car accelerates from rest to a speed of \(27 \, \mathrm{m/s}\) in \(9.0 \, \mathrm{s}\), we can calculate it using the formula:\[a_t = \frac{\Delta v}{\Delta t} = \frac{27 \, \mathrm{m/s}}{9.0 \, \mathrm{s}} = 3 \, \mathrm{m/s^2}\]

Step 2: Find the Tangential Force

The tangential force (\(F_t\)) can be found using Newton's second law \(F = ma\). Multiply the mass of the car by the tangential acceleration calculated: \[F_t = m \cdot a_t = 1150 \times 3 = 3450 \, \mathrm{N}\]

Step 3: Determine the Centripetal Acceleration

The centripetal acceleration (\(a_c\)) is found using the formula for centripetal acceleration: \[a_c = \frac{v^2}{r} = \frac{(27 \, \mathrm{m/s})^2}{450 \, \mathrm{m}} = 1.62 \, \mathrm{m/s^2}\]

Step 4: Find the Centripetal Force

The centripetal force (\(F_c\)) is also found using Newton's second law, using the centripetal acceleration:\[F_c = m \cdot a_c = 1150 \times 1.62 = 1863 \, \mathrm{N}\]

Step 5: Conclusion

The tangential component of the net force exerted on the car is \(3450 \, \mathrm{N}\), and the centripetal component is \(1863 \, \mathrm{N}\).

Key Concepts

Tangential Acceleration

Tangential acceleration refers to the rate at which an object speeds up or slows down along a curved path. It is particularly relevant when an object is in motion and changing its speed over time. Tangential acceleration occurs parallel to the direction of the velocity.

In the problem provided, the car increased its speed from rest to a higher velocity over a period of time. This change in speed allows us to calculate the tangential acceleration using the formula for acceleration: \[ a_t = \frac{\Delta v}{\Delta t} \] where \( \Delta v \) is the change in velocity and \( \Delta t \) is the change in time. Using the given values, \( a_t = \frac{27 \, \mathrm{m/s}}{9.0 \, \mathrm{s}} = 3 \, \mathrm{m/s^2} \).

Understanding tangential acceleration is crucial as it relates to how forces are applied to change the speed of an object along its path. It is always directed along the path of the motion and can be positive (speeding up) or negative (slowing down).

Centripetal Acceleration

Centripetal acceleration is the inward acceleration required for an object to move along a circular path. It keeps the object from flying outward due to its inertia, which is the tendency of an object to continue in its straight-line motion.

This acceleration is always directed toward the center of the circular path, helping objects maintain the curve. The formula for centripetal acceleration is given by: \[ a_c = \frac{v^2}{r} \] where \( v \) is the velocity, and \( r \) is the radius of the path.

Applying this formula to our problem, we substitute the given values: \( a_c = \frac{(27 \, \mathrm{m/s})^2}{450 \, \mathrm{m}} = 1.62 \, \mathrm{m/s^2} \).

Centripetal acceleration does not affect the speed of the car along the curve but ensures it stays on the path by continuously changing the direction of the velocity vector.

Newton's Second Law

Newton's Second Law is fundamental in understanding how the motion of an object changes under the influence of net forces. It states that the force on an object is equal to the mass of the object multiplied by its acceleration: \[ F = ma \] In the context of our exercise, we can use this law to find both the tangential and centripetal components of the force exerted on the car.

For the tangential force, the mass of the car is multiplied by its tangential acceleration: \[ F_t = 1150 \, \mathrm{kg} \times 3 \, \mathrm{m/s^2} = 3450 \, \mathrm{N} \] This tangential force is what changes the car's speed along its path.

Similarly, the centripetal force can be found using the centripetal acceleration: \[ F_c = 1150 \, \mathrm{kg} \times 1.62 \, \mathrm{m/s^2} = 1863 \, \mathrm{N} \] This force is crucial for maintaining the car's circular motion without altering its speed. Newton's second law provides a simple yet powerful framework for understanding how forces result in the acceleration and deceleration of objects in various environments.