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A wheel rotates without friction about a stationary horizontal axis at the center of the wheel. A constant tangential force equal to 80.0 N is applied to the rim of the wheel. The wheel has radius 0.120 m. Starting from rest, the wheel has an angular speed of 12.0 rev/s after 2.00 s. What is the moment of inertia of the wheel?

Short Answer

Expert verified
The moment of inertia of the wheel is approximately 0.254 kg m².

Step by step solution

01

Identify Given Information

We know that a tangential force of F=80.0 N is applied to a wheel with radius r=0.120 m. The final angular speed is ω=12.0 rev/s, which needs to be converted to radians per second. Since 1 revolution is 2π radians, ω=12.0×2π=24π rad/s. The wheel accelerates from rest, so initial angular speed ω0=0 rad/s, in a time t=2.00 s.
02

Calculate Angular Acceleration

The angular acceleration α can be calculated using the formula α=ωω0t. Substituting the known values, α=24π02=12π rad/s2.
03

Relate Force to Torque

Torque (τ) is related to force by the equation τ=rF. Substituting the known values: τ=0.120×80.0=9.6 Nm.
04

Relate Torque to Angular Acceleration

Torque is also related to angular acceleration and the moment of inertia I by the equation τ=Iα. Therefore, the moment of inertia I can be calculated as I=τα.
05

Calculate the Moment of Inertia

Substitute the known values of torque and angular acceleration into the equation I=τα: I=9.612π. This simplifies to I0.254 kg m2.

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Key Concepts

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

Angular Acceleration
Angular acceleration measures how quickly an object is speeding up or slowing down its rotation, similar to how linear acceleration describes changes in velocity for a moving object. In our example, the wheel starts from rest and reaches a particular angular speed in a given time.
To calculate angular acceleration, use the formula:
  • α=ωω0t
Here:
  • ω = final angular speed
  • ω0 = initial angular speed
  • t = time duration
In our problem, ω0=0 rad/s, ω=24π rad/s, and t=2 s. So, the angular acceleration is 12π rad/s².
Understanding angular acceleration helps determine how forces impact rotational motion, which is crucial for solving dynamic problems.
Torque
Torque is a measure of the rotational force applied to an object. It's the product of force and the radius at which this force is applied. Torque is what causes objects to rotate.
The formula for torque is:
  • τ=rF
Where:
  • τ = torque
  • r = radius
  • F = force applied
In the given exercise, a tangential force of 80 N is applied at the wheel’s rim of radius 0.120 m.
The torque is calculated as 9.6 Nm.
Torque plays a pivotal role in determining the moment of inertia, linking directly to how much an object resists angular acceleration when a torque is applied.
Angular Speed
Angular speed refers to how quickly an object is rotating, described by how many radians it moves through per second. Unlike linear speed, which deals with straight-line paths, angular speed is about rotation around a point or axis.
To convert revolutions per second to radians per second, use the conversion:
  • 1 revolution = 2π radians
So, if the wheel rotates at 12 revolutions per second:
  • Angular speed ω=12×2π=24π rad/s
Angular speed is vital to determining many other properties of rotating objects, including kinetic energy and moment of inertia. It’s also foundational in understanding rotational motion dynamics.
Physics Problem Solving
Approaching physics problems systematically helps simplify even the toughest challenges. This includes recognizing what the problem is asking for, identifying given data, and applying the correct physics principles and formulas.
Here are some steps to tackle physics problems:
  • Identify and list all given values and required findings from the problem statement.
  • Translate any units to ensure consistency (like converting revolutions per second to radians per second).
  • Use the relevant physics equations to connect the known variables and solve for the unknowns.
  • Keep track of units throughout the calculations to maintain accuracy.
By breaking down each calculation step-by-step, we can better understand the overall concepts, like how force impacts angular motion or how angular speed and acceleration are connected. Practicing these steps regularly enhances problem-solving skills and deepens our comprehension of physical phenomena.

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Most popular questions from this chapter

A large wooden turntable in the shape of a flat uniform disk has a radius of 2.00 m and a total mass of 120 kg. The turntable is initially rotating at 3.00 rad/s about a vertical axis through its center. Suddenly, a 70.0-kg parachutist makes a soft landing on the turntable at a point near the outer edge. (a) Find the angular speed of the turntable after the parachutist lands. (Assume that you can treat the parachutist as a particle.) (b) Compute the kinetic energy of the system before and after the parachutist lands. Why are these kinetic energies not equal?

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?

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You are designing a system for moving aluminum cylinders from the ground to a loading dock. You use a sturdy wooden ramp that is 6.00 m long and inclined at 37.0 above the horizontal. Each cylinder is fitted with a light, frictionless yoke through its center, and a light (but strong) rope is attached to the yoke. Each cylinder is uniform and has mass 460 kg and radius 0.300 m. The cylinders are pulled up the ramp by applying a constant force F to the free end of the rope. F is parallel to the surface of the ramp and exerts no torque on the cylinder. The coefficient of static friction between the ramp surface and the cylinder is 0.120. (a) What is the largest magnitude F can have so that the cylinder still rolls without slipping as it moves up the ramp? (b) If the cylinder starts from rest at the bottom of the ramp and rolls without slipping as it moves up the ramp, what is the shortest time it can take the cylinder to reach the top of the ramp?

You complain about fire safety to the landlord of your high-rise apartment building. He is willing to install an evacuation device if it is cheap and reliable, and he asks you to design it. Your proposal is to mount a large wheel (radius 0.400 m) on an axle at its center and wrap a long, light rope around the wheel, with the free end of the rope hanging just past the edge of the roof. Residents would evacuate to the roof and, one at a time, grasp the free end of the rope, step off the roof, and be lowered to the ground below. (Ignore friction at the axle.) You want a 90.0-kg person to descend with an acceleration of g/4. (a) If the wheel can be treated as a uniform disk, what mass must it have? (b) As the person descends, what is the tension in the rope?

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