Question

A ball having a mass of \(1.00 \mathrm{~kg}\) is attached to a string \(1.00 \mathrm{~m}\) long and is whirled in a vertical circle at a constant speed of \(10.0 \mathrm{~m} / \mathrm{s}\) a) Determine the tension in the string when the ball is at the top of the circle. b) Determine the tension in the string when the ball is at the bottom of the circle. c) Consider the ball at some point other than the top or bottom. What can you say about the tension in the string at this point?

Short answer

Answer: The tension force acting on the ball is 9.81 N at the top of the circle and 110.81 N at the bottom of the circle.

Step-by-step solution

Identify the forces acting on the ball when it is at the top of the circle

At the top of the circle, the two forces acting on the ball are: 1. Gravitational force (weight), \(mg\), acting vertically downward. 2. Tension force, \(T\), acting along the string towards the center. Since both forces have the same direction at the top of the circle, they are added together.

Apply Newton's second law for circular motion at the top of the circle

Newton's second law for circular motion states that the net force towards the center of the circle is equal to the mass of the object multiplied by its centripetal acceleration, \(F_{net} = ma_c\). In our case, \(F_{net} = T + mg\), and the centripetal acceleration, \(a_c = \dfrac{v^2}{r}\), where \(v\) is the constant speed of the ball and \(r\) is the radius of the circle. For the given problem, we have \(m = 1.00 kg\), \(v = 10.0 m/s\), and \(r = 1.00 m\). We can plug in these values and solve for the tension: \(T + mg = ma_c\)

Calculate tension at the top of the circle

Plugging in the values for mass, speed, and radius, we get: \(T + (1.00 kg)(9.81 m/s^2) = (1.00 kg)\left(\dfrac{(10.0 m/s)^2}{1.00 m}\right)\) After solving the equation for \(T\), we get: \(T = - 9.81 N\) However, tension cannot be negative. The negative sign indicates that our assumption about the direction of tension at the top of the circle is incorrect. Hence, \(T = 9.81 N\) at the top of the circle.

Identify the forces acting on the ball when it is at the bottom of the circle

At the bottom of the circle, the forces acting on the ball are: 1. Gravitational force (weight), \(mg\), acting vertically downward. 2. Tension force, \(T\), acting along the string towards the center. In this case, these forces have opposite directions, so the tension force will be greater than the gravitational force.

Apply Newton's second law for circular motion at the bottom of the circle

Using Newton's second law for circular motion, and noting that tension force is greater than gravitational force, we have: \(F_{net} = T - mg = ma_c\)

Calculate tension at the bottom of the circle

Plugging in the values for mass, speed, and radius, we get: \(T - (1.00 kg)(9.81 m/s^2) = (1.00 kg)\left(\dfrac{(10.0 m/s)^2}{1.00 m}\right)\) Solving this equation for \(T\), we obtain: \(T = 110.81 N\)

Discuss the tension in the string at any other point in the circle

At any point other than the top or the bottom of the circle, the tension force will have a component in the radial direction, which will contribute to the centripetal force. The gravitational force will also have radial and tangential components. Since these components would vary depending on the angle, the tension in the string will have different values at different points in the circle. In general, the tension will be greater at the bottom and decrease as the ball moves towards the top of the circle, reaching its minimum value at the top. Then, the tension will increase again as the ball moves from the top to the bottom of the circle.

Key concepts

Newton's Second Law

Newton's Second Law is a fundamental principle that describes how the velocity of an object changes when it is subject to an external force. In simple terms, it tells us that the force acting on an object is equal to the mass of the object multiplied by its acceleration, expressed in the equation:
\[ F = ma \]
For objects in circular motion, like our ball on a string, we apply Newton's Second Law to determine the centripetal forces required to keep the object moving in a circle. In the case of circular motion, the acceleration component is the centripetal acceleration \(a_c\), which keeps the object moving along a curved path. It is calculated using the formula:
\[ a_c = \frac{v^2}{r} \]
where \(v\) is the velocity and \(r\) is the radius of the circle. The net force in the direction of the center of the circle becomes crucial. At the top of the circle, both gravity and tension act towards the center, contributing to the required net centripetal force. Hence, Newton's Second Law is the basis for calculating the tension in the string at different positions along the circular path.

Centripetal Force

Centripetal Force is what keeps an object moving in a circle rather than in a straight line. It's an inward force, towards the center of the circle, preventing the object from flying off tangentially due to inertia. This force can have different sources depending on the setup of the system.
- For our ball in vertical motion, the centripetal force is generated by both the tension in the string and the gravitational pull. - At the top of the circle, the gravitational force \(mg\) and tension \(T\) contribute together towards the required centripetal force. The equation becomes:
\[ T + mg = m \frac{v^2}{r} \]This sums the forces to produce the needed centripetal acceleration.
- At the bottom, tension works oppositely to gravity requiring:
\[ T - mg = m \frac{v^2}{r} \]Here, tension has to not only compensate for the gravitational pull but also provide the required centripetal force by itself.
This balance of forces ensures the ball moves in a circle, underscoring the relevance of centripetal force in maintaining circular motion.

Tension in Strings

Tension in Strings arises from the need to maintain the circular motion of the object, acting as the force that pulls the object towards the center of the circle. The tension varies based on the position of the ball in the circle due to different contributions from gravitational forces.
- **At the top of the circle:** Tension and the gravitational force act in the same direction. This means both help to provide the centripetal force necessary to keep the ball moving in its path. As a result, the string bears a smaller force by itself, due to the help from gravity. - **At the bottom of the circle:** The tension in the string has to counteract the gravitational force which acts against it. Here, the string must pull harder, resulting in a larger tension, to keep the ball from deviating from its path.
- **Other points in the circle:** The tension will change based on the angle of the ball's position as both radial and tangential components of gravitational force vary, making the tension dynamic throughout the circular path.
In essence, tension provides the necessary force for centripetal motion, adapting itself as required based on positional changes of the object within the circle.