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Chapter 5: Problem 49

In a skating stunt known as crack-the-whip, a number of skaters hold hands and form a straight line. They try to skate so that the line rotates about the skater at one end, who acts as the pivot. The skater farthest out has a mass of 80.0 kg and is 6.10 m from the pivot. He is skating at a speed of 6.80 m/s. Determine the magnitude of the centripetal force that acts on him.

Short Answer

Expert verified
The centripetal force is approximately 606.42 N.

Step by step solution

01

Identify the relevant formula

The centripetal force acting on an object moving in a circle is given by the formula:\[ F_c = \frac{mv^2}{r} \]where \( m \) is the mass of the object, \( v \) is the tangential speed, and \( r \) is the radius of the circular path.
02

Substitute the known values

In this problem, the mass \( m \) of the skater is given as 80.0 kg, the radius \( r \) is 6.10 m (distance from the pivot), and the speed \( v \) is 6.80 m/s. Substitute these values into the formula:\[ F_c = \frac{80.0 \times (6.80)^2}{6.10} \]
03

Calculate the force

First, calculate the numerator:\( 80.0 \times (6.80)^2 = 80.0 \times 46.24 \). This results in \( 3699.2 \).Next, divide by the radius:\[ F_c = \frac{3699.2}{6.10} \approx 606.42 \]Therefore, the centripetal force is approximately 606.42 N.

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Key Concepts

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

Circular Motion
Circular motion is the movement of an object along the circumference of a circle or rotation along a circular path. When an object undergoes circular motion, it's continuously changing direction, which means it is accelerating鈥攅ven if it's moving at constant speed. This is due to centripetal acceleration, which points towards the center of the circle.
  • Centripetal Force: It is this center-seeking force that keeps an object moving in a circle. Without it, an object would move off in a straight line due to inertia.
  • Velocity in Circular Motion: While speed could be constant, velocity is not. This is because velocity is a vector quantity, meaning it has both magnitude and direction. In circular motion, the direction is constantly changing.
  • Examples: Planets orbiting the sun, cars turning in a circle, and of course, the crack-the-whip skating motion where skaters rotate about a central point.
Understanding circular motion is important as it is a fundamental concept in physics that explains how objects behave when they are constrained to follow a curved path rather than moving linearly.
Crack-the-whip
"Crack-the-whip" is a skating game that visually demonstrates the principles of circular motion in a real-world context. In this activity, skaters hold hands and rotate around a central pivot skater. The skater at the end of the chain travels at the highest speed and experiences the greatest force.
  • Pivotal Point: The person closest to the center acts as a pivot and moves slowly compared to others at the end.
  • Increased Speed and Force: As you move outward, each skater needs to cover more distance in the same amount of time, so their speed increases along with the centripetal force acting on them.
  • Concept in Action: This creates a visual and tangible way to understand how centripetal forces act in circular motion, as the skater on the end must resist flying outward through centripetal force.
This physics-based game illustrates the connection between rotational speed and force, aiding in grasping more abstract physics concepts.
Physics Problem-Solving
Solving physics problems requires a structured approach to ensure every step is logical and leads to the correct solution. For understanding centripetal force problems like the example of crack-the-whip, follow these steps:
  • Identify the Problem: Read through and understand what is being asked. For the crack-the-whip, identify the outer skater's mass, the radius of motion, and speed.
  • Use the Right Formula: Identify and write down the formula that applies, such as the centripetal force formula, \( F_c = \frac{mv^2}{r} \).
  • Substitute Values: Substitute the given values logically into the formula.
  • Calculate and Confirm: Perform the arithmetic, keeping an eye on units, to find the final answer. Double-check calculations for accuracy.
  • Contextual Understanding: Finally, put the answer in the context of the physical situation, ensuring it makes sense within the problem.
Following this structured approach makes complex physics problems more approachable and less intimidating.
Rotational Dynamics
Rotational dynamics involves studying motion where forces cause objects to rotate. It connects directly to circular motion, providing further depth in understanding how objects move under constraints with forces involved.
  • Torque: It's the measure that causes an object to rotate. In the crack-the-whip scenario, torque is applied to keep the skaters rotating around their pivot.
  • Angular Velocity: This represents the rate of rotation and is crucial in understanding how speed changes as you move further from the pivot.
  • Moment of Inertia: Similar to mass in linear dynamics, it's the rotational equivalent that describes how difficult it is to change an object's rotational state.
  • Real-world Application: Rotational dynamics are not just theoretical; they apply to everything from machinery and vehicle wheels to sports and playground physics.
Understanding rotational dynamics fosters a deeper appreciation of how rotational forces are managed in everyday life, and how they interplay with linear dynamics to produce complex motion patterns.

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Most popular questions from this chapter

A car travels at a constant speed around a circular track whose radius is 2.6 km. The car goes once around the track in 360 s. What is the magnitude of the centripetal acceleration of the car?

A car is safely negotiating an unbanked circular turn at a speed of 21 m/s. The road is dry, and the maximum static frictional force acts on the tires. Suddenly a long wet patch in the road decreases the maximum static frictional force to one-third of its dry-road value. If the car is to continue safely around the curve, to what speed must the driver slow the car?

A centrifuge is a device in which a small container of material is rotated at a high speed on a circular path. Such a device is used in medical laboratories, for instance, to cause the more dense red blood cells to settle through the less dense blood serum and collect at the bottom of the container. Suppose the centripetal acceleration of the sample is \(6.25 \times 10^{3}\) times as large as the acceleration due to gravity. How many revolutions per minute is the sample making, if it is located at a radius of 5.00 \(\mathrm{cm}\) from the axis of rotation?

The earth rotates once per day about an axis passing through the north and south poles, an axis that is perpendicular to the plane of the equator. Assuming the earth is a sphere with a radius of \(6.38 \times 10^{6}\) m, determine the speed and centripetal acceleration of a person situated (a) at the equator and (b) at a latitude of 30.0 north of the equator.

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