Ch 10: Dynamics of Rotational Motion

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 10: Dynamics of Rotational Motion

Problem 34

Chapter 10, Problem 34

An electric motor consumes 9.00 kJ of electrical energy in 1.00 min. If one-third of this energy goes into heat and other forms of internal energy of the motor, with the rest going to the motor output, how much torque will this engine develop if you run it at 2500 rpm?

Verified step by step guidance

First, convert the electrical energy consumed from kilojoules to joules. Since 1 kJ = 1000 J, the motor consumes 9.00 kJ * 1000 = 9000 J of energy.

Determine the energy that is converted into useful work. Since one-third of the energy is lost to heat and other forms of internal energy, two-thirds of the energy is used for work. Calculate this as (2/3) * 9000 J.

Convert the motor's rotational speed from revolutions per minute (rpm) to radians per second. Use the conversion factor: 1 rpm = (2π/60) rad/s. Therefore, 2500 rpm = 2500 * (2π/60) rad/s.

Use the formula for power in terms of torque and angular velocity: P = τ * ω, where P is power, τ is torque, and ω is angular velocity. Rearrange this formula to solve for torque: τ = P / ω.

Calculate the power output of the motor using the energy used for work and the time duration. Power is energy divided by time, so P = (energy for work) / (1 min converted to seconds). Then, substitute the values of power and angular velocity into the torque formula to find the torque.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Energy Conversion

Energy conversion in an electric motor involves transforming electrical energy into mechanical energy. In this scenario, the motor consumes 9.00 kJ of electrical energy, with one-third lost to heat and internal energy. Understanding how energy is distributed between useful work and losses is crucial for calculating the motor's output.

Torque

Torque is a measure of the rotational force produced by the motor. It is calculated using the formula: Torque = Power / Angular Velocity. Knowing the power output and the motor's rotational speed (2500 rpm) allows us to determine the torque, which is essential for understanding the motor's mechanical performance.

Angular Velocity

Angular velocity refers to the rate of rotation, measured in revolutions per minute (rpm) or radians per second. In this problem, the motor runs at 2500 rpm, which must be converted to radians per second for torque calculations. This conversion is vital for applying the torque formula accurately.

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