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Chapter 7: Problem 21

A \(55.0-\mathrm{kg}\) ice skater is moving at \(4.00 \mathrm{~m} / \mathrm{s}\) when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius \(0.800 \mathrm{~m}\) around the pole. (a) Determine the force exerted by the horizontal rope on her arms. (b) Compare this force with her weight.

Short Answer

Expert verified
The force exerted by the horizontal rope on her arms is 1100 N, which is about two times her weight(539 N).

Step by step solution

01

Calculate the Centripetal Force

To calculate the force exerted by the horizontal rope on her arms, we need to use the formula for the centripetal force, which is \( F_c = m \cdot v^2 / r \) , where \( m \) is the skater鈥檚 mass, \( v \) is her speed, and \( r \) is the radius of the circle. So, \( F_c = 55.0kg \cdot (4.00 m/s)^2 / 0.800m = 1100 N \) . The force exerted by the rope on her arms is therefore 1100 N.
02

Calculate the Skater's Weight

Weight is calculated by the formula \( W = m \cdot g \) , where \( m \) is the mass and \( g \) is the acceleration due to gravity, which is approximately \( 9.8 m/s^2 \). Therefore, \( W = 55.0kg \cdot 9.8 m/s^2 = 539 N \) .
03

Compare the Centripetal Force with the Skater's Weight

We have found that the force exerted by the rope, or the centripetal force, is 1100 N. Meanwhile, the skater's weight is 539 N. By comparing these two values, we can see that the force exerted by the rope on her arms is about two times her weight

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Key Concepts

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

Circular Motion
Circular motion occurs when an object moves in a path that forms a circle. In the case of the ice skater, she grabs a rope and travels in a circular path around a pole. This type of motion is governed by the concept of centripetal force, which is essential for keeping the object moving along the circular path.

The centripetal force is a "center-seeking" force. It acts perpendicular to the motion of the object, pulling it towards the center of the circular path. In our example, the rope provides this force, allowing the skater to continue her circular trajectory. Without this force, the skater would move off in a straight line due to inertia, a concept from Newton's first law of motion.
  • Key to understanding circular motion is recognizing that the velocity of the object is constantly changing direction, even if its speed remains constant.
  • This change in direction is accelerated by the centripetal force.
Newton's Laws of Motion
Newton's laws of motion are fundamental to understanding mechanics, including circular motion. Let's explore how these laws relate to the problem of the ice skater.

  • **Newton's First Law:** A body at rest will remain at rest, and a body in motion will remain in motion at a constant velocity unless acted upon by a force. For the skater, without the force exerted by the rope (the centripetal force), she would move in a straight line due to inertia.
  • **Newton's Second Law:** The acceleration of an object is directly proportional to the net force acting upon the object and inversely proportional to its mass. The formula is: \( F = m imes a \) Here, the first and second steps show us the importance of this law. The tension in the rope provides the centripetal force necessary to change the direction of her motion, resulting in circular motion.
  • **Newton's Third Law:** For every action, there is an equal and opposite reaction. The skater exerts a force on the rope, and therefore, the rope exerts an equal and opposite centripetal force on the skater's arms.
Gravitational Force
Gravitational force is another key concept and it affects every object with mass. It pulls objects towards the center of the Earth, and is very evident in the weight of the skater. The skater's weight is a result of the gravitational force acting on her mass.

The formula for calculating weight due to gravity is: \( W = m \cdot g \)where
  • \( m \) is the mass of the skater,
  • \( g \) is the acceleration due to gravity, which is approximately \( 9.8\ m/s^2 \).
In comparing the two forces acting on the skater - the centripetal force of 1100 N and her weight of 539 N - it's clear that the force needed for her circular motion is much greater than the gravitational pull. This illustrates how significant the centripetal force is in maintaining her movement around the pole. Without the sufficient centripetal force, her motion would not be sustained in a circular path.

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Most popular questions from this chapter

A sample of blood is placed in a centrifuge of radius \(15.0 \mathrm{~cm}\). The mass of a red blood cell is \(3.0 \times 10^{-16} \mathrm{~kg}\), and the magnitude of the force acting on it as it settles out of the plasma is \(4.0 \times 10^{-11} \mathrm{~N}\). At how many revolutions per second should the centrifuge be operated?

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