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Chapter 5: Problem 19

IP A 71 -kg parent and a \(19-k g\) child meet at the center of an ce rink. They place their hands together and push. (a) Is the force experienced by the child more than, less than, or the same as the force experienced by the parent? (b) Is the acceleration of the child more than, less than, or the same as the acceleration of the parent? Explain. (c) If the acceleration of the child is \(2.6 \mathrm{m} / \mathrm{s}^{2}\) in magnitude, what is the magnitude of the parent's ncceleration?

Short Answer

Expert verified
(a) The force is the same. (b) The child's acceleration is more. (c) Parent's acceleration is 0.695 m/s虏.

Step by step solution

01

Understanding Forces

According to Newton's Third Law of Motion, when two people push against each other, the forces they exert on each other are equal and opposite. Therefore, the force experienced by the child is the same as the force experienced by the parent. The answer to part (a) is that the force is the same.
02

Comparing Accelerations

Acceleration can be determined using Newton's Second Law, expressed as \( F = ma \). Although the forces are equal, the masses of the parent and child are different, which affects their acceleration. Thus, \( a = \frac{F}{m} \). Since the child's mass is less, the acceleration of the child is more than that of the parent because \( F = m_{child} imes a_{child} = m_{parent} imes a_{parent} \). The answer to part (b) is that the acceleration of the child is more.
03

Calculating Parent's Acceleration

We know that the same force \( F \) acts on both the parent and the child but their masses are different. Given the child's acceleration \( a_{child} = 2.6 \text{ m/s}^2 \) and mass \( m_{child} = 19 \text{ kg} \), we find \( F \) as \( F = m_{child} \times a_{child} = 19 \times 2.6 = 49.4 \text{ N} \). The parent's acceleration \( a_{parent} \) can be determined by rearranging the formula for force (\( F = m_{parent} \times a_{parent} \)), giving \( a_{parent} = \frac{F}{m_{parent}} = \frac{49.4}{71} = 0.695 \text{ m/s}^2 \). The magnitude of the parent's acceleration is approximately \(0.695 \text{ m/s}^2\).

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Key Concepts

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

Newton's Third Law of Motion
Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction.
This means when two people push against each other, like the parent and child in the ice rink example, they exert forces on each other that are equal in magnitude but opposite in direction.
Thus, despite their size difference, the force each experiences is the same. Here鈥檚 how it works in simple terms:
  • The child pushes against the parent, and the parent pushes back with an equal force.
  • This force is not dependent on who is stronger or bigger.
  • The forces cancel each other out as far as the interaction goes, maintaining balance.
This principle helps us understand why the force felt by both individuals is equal, simplifying what might initially appear as a complex scenario.
Newton's Second Law of Motion
Newton's Second Law of Motion links force, mass, and acceleration through the equation \( F = ma \).
It tells us that the force on an object is equal to the mass of the object multiplied by the acceleration it experiences.
In the context of the parent and child on the ice, while the forces are equal, their masses are different, causing different accelerations.Breaking it down:
  • Acceleration \( (a) \) depends on both the force \( (F) \) applied and the mass \( (m) \) of the object.
  • The child, having less mass, will experience a greater acceleration with the same force.
  • Conversely, the larger mass of the parent results in a smaller acceleration.
This understanding explains why under the same force, the child accelerates faster than the parent.
Force and Acceleration Dynamics
The relationship between force and acceleration is crucial in predicting motion, as seen with the parent and child on the ice.
Given the child's known acceleration and mass, we can calculate the force using \( F = m_{child} \times a_{child} \).
With the force calculated, this same force is applied to determine the parent's acceleration using their mass.Steps in calculation:
  • First, compute the force using the child's metrics: \( F = 19 \times 2.6 = 49.4 \text{ N} \).
  • Use this force to find the parent's acceleration: \( a_{parent} = \frac{49.4}{71} \approx 0.695 \text{ m/s}^2 \).
This illustrates how acceleration varies with mass even under identical forces, reinforcing key dynamics between force, mass, and acceleration.

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Most popular questions from this chapter

IP B I O The Force of Running Biomechanical research has shown that when a \(67-\mathrm{kg}\) person is running, the force exerted on each foot as it strikes the ground can be as great as \(2300 \mathrm{N}\). (a) What is the ratio of the force exerted on the foot by the ground to the person's body weight? (b) If the only forces acting on the person are (i) the force exerted by the ground and (ii) the person's weight, what are the magnitude and direction of the person's acceleration? (c) If the acceleration found in part (b) acts for \(10.0 \mathrm{ms},\) what is the resulting change in the vertical component of the person's velocity?

\(\cdot\) CE A skateboarder on a ramp is accelerated by a nonzero net force. For each of the following statements, state whether it is always true, never true, or sometimes true. (a) The skateboarder is moving in the direction of the net force. (b) The ac- celeration of the skatcboarder is at right angles to the net force. (c) The acceleration of the skateboarder is in the same direction as the net force. (d) The skateboarder is instantaneously at rest.

Two boxes are at rest on a smooth, horizontal surface. The boxes are in contact with one another. If box 1 is pushed with a force of magnitude \(F=12.00 \mathrm{N}\), the contact force between the boxes is \(8.50 \mathrm{N} ;\) if, instead, box 2 is pushed with the force \(F\) the contact force is \(12.00 \mathrm{N}-8.50 \mathrm{N}=3.50 \mathrm{N}\). In either case, the boxes move together with an acceleration of \(1.70 \mathrm{m} / \mathrm{s}^{2}\). What is the mass of (a) box 1 and (b) box \(2 ?\)

(a) Find the normal force exerted on a \(2.9-\mathrm{kg}\) book resting on a surface inclined at \(36^{\circ}\) above the horizontal. (b) If the angle of the incline is reduced, do you expect the normal force to increase, decrease, or stay the same? Explain.

Predict/Explain You jump out of an airplane and open your parachute after an extended period of free fall. (a) To decelerate your fall, must the force exerted on you by the parachute be greater than, less than, or equal to your weight? (b) Choose the best explanation from among the following: I. Parachutes can only exert forces that are less than the weight of the skydiver. II. The parachute exerts a force exactly equal to the skydiver's weight. III. To decelerate after free fall, the net force acting on a skydiver must be upward.
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