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Chapter 10: Problem 4

\(A 4 N\) and a 10 N force act on an object. The moment arm of the \(4 \mathrm{~N}\) force is \(0.2 \mathrm{~m}\). If the \(10 \mathrm{~N}\) force produces five times the torque of the \(4 \mathrm{~N}\) force, what is its moment arm?

Short Answer

Expert verified
The moment arm of the 10 N force is 0.4 m.

Step by step solution

01

Understanding Torque

Torque \( \tau \) is a measure of the force causing an object to rotate. It is calculated by the formula \( \tau = F \times r \), where \( F \) is the force applied and \( r \) is the moment arm or the perpendicular distance from the pivot point to the line of action of the force.
02

Calculate Torque of 4 N Force

The torque produced by the 4 N force is \( \tau_1 = 4 \text{ N} \times 0.2 \text{ m} = 0.8 \text{ Nm} \).
03

Set Up Equation for 10 N Force

The problem states that the 10 N force produces five times the torque of the 4 N force. Therefore, \( \tau_2 = 5 \times \tau_1 = 5 \times 0.8 \text{ Nm} = 4.0 \text{ Nm} \).
04

Solve for the Moment Arm of 10 N Force

Using \( \tau_2 = F \times r \), where \( F = 10 \text{ N} \), we have \( 4.0 \text{ Nm} = 10 \text{ N} \times r \). Solving for \( r \): \( r = \frac{4.0 \text{ Nm}}{10 \text{ N}} = 0.4 \text{ m} \).

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Key Concepts

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

Moment Arm
The moment arm is the perpendicular distance between the line of action of a force and the axis of rotation. It plays a crucial role in determining the amount of torque a force can exert. Think of it like a lever arm that magnifies the force's effect in causing rotational motion. The longer the moment arm, the greater the torque for the same force magnitude.
  • Example: If you are pushing a door open, the handle is located at the end of the door, maximizing the moment arm and making it easier to open with the same effort.
  • In physics problems, correctly identifying and calculating the moment arm is key to solving for torque and understanding rotational dynamics.
In the context of the exercise, the moment arm for the 4 N force is given as 0.2 m, which is used to calculate the torque.
Force
Force is a vector quantity that represents an interaction that, when unopposed, will change the motion of an object. It is not only about how much you push or pull; the direction also matters. In rotational dynamics, the force's position relative to the pivot point is crucial in defining its effectiveness.
  • The standard unit of force in the International System of Units (SI) is the Newton (N).
  • Different forces can act simultaneously, as seen in the exercise with a 4 N and a 10 N force.
The exercise demonstrates how varying forces and their application points (moment arms) result in different torques.
Rotational Dynamics
Rotational dynamics studies the motion of objects that rotate about an axis. It parallels linear dynamics but instead of dealing with linear acceleration and force, we consider angular acceleration and torque. This branch of physics describes how objects spin, turn, or roll and how forces exert torques that change their rotational motion.
  • Torque, analogous to force in linear dynamics, is essential in initiating or stopping rotational motion.
  • In the exercise, you need to calculate torque to understand the rotational effects of the forces involved.
Solving problems in rotational dynamics requires understanding the relationship between force, moment arm, and torque.
Physics Problem Solving
Problem solving in physics involves using a systematic approach to understand and solve exercises. Breaking down problems into smaller, manageable parts is key to success in this area. Here鈥檚 a simple approach you can follow:
Identify the knowns and unknowns: List out all givens from the problem statement, such as forces and distances.
  • Apply relevant formulas: Use formulas that relate to the concepts involved, such as torque \( \tau = F \times r \).
  • Solve algebraically: Manipulate equations to solve for the unknowns, ensuring you follow mathematical rules correctly.
  • Check your solution: Review your work to confirm that the solution makes sense both mathematically and conceptually.
In this exercise, understanding and applying these steps enabled us to find the moment arm for the 10 N force effectively.

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Most popular questions from this chapter

A large turntable rotates about a fixed vertical axis, making one revolution in 6.00 s. The moment of inertia of the turntable about this axis is \(1200 \mathrm{~kg} \cdot \mathrm{m}^{2}\). A child of mass \(40.0 \mathrm{~kg}\). initially standing at the center of the turntable, runs out along a radius. What is the angular speed of the turntable when the child is \(2.00 \mathrm{~m}\) from the center, assuming that you can treat the child as a particle?

A therapist tells a \(74 \mathrm{~kg}\) patient with a broken leg that he must have his leg in a cast suspended horizontally. For minimum discomfort, the leg should be supported by a vertical strap attached at the center of mass of the leg-cast system. (See Figure \(10.62 .)\) In order to comply with these instructions, the patient consults a table of typical mass distributions and finds that both upper legs (thighs) together typically account for \(21.5 \%\) of body weight and the center of mass of each thigh is \(18.0 \mathrm{~cm}\) from the hip joint. The patient also reads that the two lower legs (including the feet) are \(14.0 \%\) of body weight, with a center of mass \(69.0 \mathrm{~cm}\) from the hip joint. The cast has a mass of \(5.50 \mathrm{~kg}\), and its center of mass is 78.0 \(\mathrm{cm}\) from the hip joint. How far from the hip joint should the supporting strap be attached to the cast?

Two people carry a heavy electric motor by placing it on a light board \(2.00 \mathrm{~m}\) long. One person lifts at one end with a force of \(400.0 \mathrm{~N}\), and the other lifts at the opposite end with a force of \(600.0 \mathrm{~N}\). (a) Start by making a free-body diagram of the motor. (b) What is the weight of the motor? (c) Where along the board is its center of gravity located?

His body is again at \(30.0^{\circ}\) to the vertical, but now the height at which the rope is held above- but still parallel to the ground-is varied. The tension in the rope in front of the competitor \(\left(T_{1}\right)\) is measured as a function of the shortest distance between the rope and the ground (the holding height). Tension \(T_{1}\) is found to decrease as the holding height increases. What could explain this observation? As the holding height increases. A. the moment arm of the rope about his feet decreases due to the angle that his body makes with the vertical. B. the moment arm of the weight about his feet decreases due to the angle that his body makes with the vertical. C. a smaller tension in the rope is needed to produce a torque sufficient to balance the torque of the weight about his feet. D. his center of mass moves down to compensate, so less tension in the rope is required to maintain equilibrium.

A cord is wrapped around the rim of a wheel \(0.250 \mathrm{~m}\) in radius, and a steady pull of \(40.0 \mathrm{~N}\) is exerted on the cord. The wheel is mounted on frictionless bearings on a horizontal shaft through its center. The moment of incrtia of the wheel about this shaft is \(5.00 \mathrm{~kg} \cdot \mathrm{m}^{2}\). Compute the angular acceleration of the wheel.
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