Question

A stone weighing 1 pound is attached to the end of a two-foot string and is whirled horizontally with one end held fixed. It makes 1 revolution per second. Find the work done by the force \(\mathbf{F}\) that keeps the stone moving in a circular path. [Hint: Use Force \(=\) (mass)(centripetal acceleration).]

Short answer

To solve this problem, first the speed of the stone was found using the formula for the circumference of a circle. This speed was then used to calculate the centripetal acceleration. The resulting acceleration multiplied by the mass of the stone gives the force keeping the stone in circular motion. Multiplying this force by the distance travelled (the circumference of the circle) gives the work done by the force.

Step-by-step solution

Calculate the Speed of the Stone

The speed of the stone is given by the circumference of the circle it travels divided by the time taken. Since it completes one revolution in a second, the speed, \(v\), is calculated as \(v = 2\pi r\), where \(r = 0.6096m\) is the radius of the circular path.

Determine the Centripetal Acceleration

The centripetal acceleration, \(a\), is given by \(a = \frac{v^2}{r}\), where \(v\) is the speed calculated in Step 1, and \(r = 0.6096m\) is the radius of the circular path. Substitute the values into the formula to obtain \(a\).

Calculate the Force

The force, \(\mathbf{F}\), keeping the stone in circular motion is given by the formula, \(\mathbf{F} = ma\). The mass, \(m = 0.4536 kg\), and \(a\) is the centripetal acceleration, which was calculated in Step 2. Multiply \(m\) and \(a\) to find force, \(\mathbf{F}\).

Calculate the Work Done

The work done by a force over a distance is given by the product of force and distance. In this case, since the stone completes one revolution, the distance, \(d\), is the circumference of the circular path, which equals \(2\pi r\). The work done, \(W\), is therefore calculated as \(W = \mathbf{F} d\), substituting \(\mathbf{F}\) from Step 3 and \(d = 2\pi r\).

Key concepts

Centripetal Acceleration

When an object moves in a circle, it constantly changes direction, requiring an inward force to keep it moving along that curved path. This is where centripetal acceleration comes into play. It is the rate of change of the velocity of the object toward the center of the circular path. The formula for calculating centripetal acceleration is given by \( a = \frac{v^2}{r} \), where \( v \) is the velocity of the object and \( r \) is the radius of the circle.

Understanding centripetal acceleration is crucial because it helps explain why objects performing circular motion, like the stone in our exercise, require a force to maintain their path of motion. Without this force, the object would move off in a straight line due to inertia.

Circular Motion

Circular motion refers to the movement of an object along the circumference of a circle. In the textbook exercise, the stone attached to the string undergoes circular motion as it is whirled around. The key point about such motion is that it's always changing direction, making it a type of accelerated motion even if the speed is constant.

Revolutions and Speed

In our example, the stone makes one complete revolution per second, providing a clear way to measure its speed. To calculate this, you multiply the number of revolutions by the circumference of the circular path. For objects in circular motion, the velocity vector is always tangent to the circle, and the acceleration vector is directed inward, towards the center.

Work Done in Physics

Work in physics is the measure of energy transfer when a force applied to an object moves that object. The key to calculating work is recognizing that the object must move in the direction of the force. The fundamental formula for work (\( W \) is force (\( F \) multiplied by distance (\( d \)) or \( W = Fd \).

However, in the case of circular motion like our exercise, even though the force is constantly applied to the stone, the direction of this force is towards the center of the circular path, not along the path of motion. This means that technically, no work is done as per the physics definition because there is no displacement in the direction of the force.

Force and Motion

The relationship between force and motion is foundational in physics. Newton's laws of motion describe how forces affect the movement of objects. The force keeping an object moving in a circular path is called centripetal force, as seen with the stone exercise. It's crucial to note that the force itself does not do work in the case of pure circular motion since the motion is perpendicular to the direction of the force.

For the stone tied to a string, the tension in the string provides the centripetal force necessary to keep the stone moving in its circular path. This emphasizes the essential interplay between force and velocity in creating and maintaining motion along a curved trajectory.