Problem 71 - University Physics with Modern Physics

Question

A small block with mass 0.0500 kg slides in a vertical circle of radius \(R =\) 0.800 m on the inside of a circular track. There is no friction between the track and the block. At the bottom of the block's path, the normal force the track exerts on the block has magnitude 3.40 N. What is the magnitude of the normal force that the track exerts on the block when it is at the top of its path?

Step-by-Step Solution

Answer

The normal force at the top is approximately 2.42 N.

Step 1: Understand the Forces at the Bottom

At the bottom of the path, the normal force and the gravitational force both act on the block. The gravitational force is given by \( F_g = mg \). Therefore, the relationship for centripetal force \( F_c \) at the bottom is \( F_c = F_n - F_g \), where \( F_n = 3.40 \text{ N} \). So, we have:\[ F_c = 3.40 - 0.0500 \times 9.8 \]

Step 2: Calculate the Speed at the Bottom

Utilizing the formula for centripetal force \( F_c = \frac{mv^2}{R} \) and the result from Step 1, we can solve for the speed \( v \) at the bottom of the path:\[ \frac{mv^2}{R} = 3.40 - 0.490 \] \[ \frac{0.0500 \times v^2}{0.800} = 2.91 \] \[ v^2 = \frac{2.91 \times 0.800}{0.0500} \] \[ v^2 = 46.56 \] \[ v = \sqrt{46.56} \approx 6.82 \, \text{m/s} \]

Step 3: Analyze Forces at the Top

At the top of the path, both the gravitational force and the normal force act towards the center of the circle. This means the centripetal force is the sum of the gravitational force and the normal force: \[ F_c = F_n + F_g \]Rearrange the equation to find \( F_n \):\[ F_n = F_c - F_g \]

Step 4: Calculate Centripetal Force at the Top

The speed at the top is the same as at the bottom, 6.82 m/s, since there is no friction and energy is conserved. The centripetal force at the top using this speed is:\[ F_c = \frac{mv^2}{R} = \frac{0.0500 \times (6.82)^2}{0.800} \approx 2.910 \text{ N} \]

Step 5: Compute the Normal Force at the Top

Now, substitute \( F_c \) and \( F_g \) into the equation from Step 3:\[ F_g = mg = 0.0500 \times 9.8 = 0.490 \text{ N} \]\[ F_n = 2.910 - 0.490 \approx 2.42 \text{ N} \]

Step 6: Conclusion

The magnitude of the normal force at the top of the block's path is approximately 2.42 N.

Key Concepts

Centripetal Force

Centripetal force is an important concept in understanding motion in a circle. It is the force required to keep an object moving in a circular path and is directed towards the center of the circle. In our exercise, the small block travels in a vertical circular path. Whether it is at the top or bottom, centripetal force is essential for maintaining its circular motion.
The centripetal force (F_c) can be expressed by the formula:\[ F_c = \frac{mv^2}{R} \]where \( m \) is the mass of the object, \( v \) is the velocity, and \( R \) is the radius of the circle.
At the bottom of the path, this force is affected by the normal force and gravitational force. Meanwhile, at the top, the centripetal force is the sum of the gravitational force and normal force. These varying conditions result in different magnitudes for the normal force exerted by the track.
Understanding how centripetal force acts in different parts of the circular path is essential for solving problems involving motion in a circle.

Gravitational Force

Gravitational force is the force exerted by the Earth that pulls objects towards its center. It is experienced as weight, and is crucial in problems involving vertical circular motion like the one in our exercise. This force can be calculated using the equation:\[ F_g = mg \]Where:

• \( F_g \) is the gravitational force.
• \( m \) is the mass of the object (0.0500 kg in our case).
• \( g \) is the acceleration due to gravity, approximately 9.8 m/s² on Earth.

In the exercise, gravitational force interacts with the centripetal force to determine the normal force. At the bottom of the path, gravitational force works opposite to the normal force. At the top, it contributes towards the total centripetal force needed to maintain circular motion. Hence, understanding gravitational force's role helps us predict how other forces will vary in different positions of circular motion.

Conservation of Energy

Conservation of energy is a fundamental concept stating that energy cannot be created or destroyed in an isolated system. It can only change forms. In the exercise, the small block experiences no friction, meaning mechanical energy is conserved as it moves along the track.
The energy involved includes kinetic energy, which depends on the block's speed, and potential energy that depends on its height in a gravitational field. At the top and bottom, these energies balance each other to ensure that the total energy remains constant. The equation typically used is:\[ E_{total} = E_k + E_p \]Where:

• \( E_k = \frac{1}{2}mv^2 \) is kinetic energy.
• \( E_p = mgh \) is potential energy.

This principle helps us deduce that the speed at the top and bottom remains constant as energy transitions between kinetic and potential forms. In our exercise, this consistency allows us to compute the forces acting on the block based on a constant speed.