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Chapter 11: Problem 13

A car is headed north at \(70 \mathrm{km} / \mathrm{h}\). Give the magnitude and direction of the angular velocity of its 62 -cm-diameter wheels.

Short Answer

Expert verified
The magnitude of angular velocity of the car's wheels is 62.71 rad/s and the direction is negative (to the left of the car).

Step by step solution

01

Convert speed to standard units

The speed of the car is given in km/h. This is not the standard unit of speed, so convert this to m/s using the conversion \(1 \mathrm{km/h} = \frac{5}{18} \mathrm{m/s}\). So, \(v = 70 \mathrm{km/h} = 70 \times \frac{5}{18} \mathrm{m/s} = 19.44 \mathrm{m/s}\).
02

Determine the radius of the wheel

The diameter of the wheels is given as 62 cm. The radius is half the diameter, so \(\mathrm{r} = \frac{62}{2} \mathrm{cm} = 31 \mathrm{cm}\). Convert this to meters by dividing by 100, so \(\mathrm{r} = 31 \times \frac{1}{100} \mathrm{m} = 0.31 \mathrm{m}\).
03

Calculate the angular velocity

Now use the formula \(\omega = \frac{v}{r}\) to calculate the angular velocity. Substituting the values we've found for \(v\) and \(r\), \(\omega = \frac{19.44 \mathrm{m/s}}{0.31 \mathrm{m}} = 62.71 \mathrm{rad/s}\).
04

Determine the direction of the angular velocity

The direction of the angular velocity is determined by the right-hand rule. Wrap your right hand around the wheel with your fingers pointing in the direction of rotation (counter-clockwise when viewed from the right side of the car). Your thumb points in the direction of the angular velocity, which is towards the left of the car. This direction is generally regarded as negative, so the angular direction is negative. The magnitude remains the same.

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Most popular questions from this chapter

A particle of mass \(m\) moves in a straight line at constant speed \(v\) Show that its angular momentum about a point located a perpendicular distance \(b\) from its line of motion is mvb regardless of where the particle is on the line.

A time-dependent torque given by \(\tau=a+b\) sin \(c t\) is applied to an object that's initially stationary but is free to rotate. Here \(a, b\) and \(c\) are constants. Find an expression for the object's angular momentum as a function of time, assuming the torque is first applied at \(t=0\).

A baseball player extends his arm straight up to catch a \(145-\mathrm{g}\) baseball moving horizontally at \(42 \mathrm{m} / \mathrm{s}\). It's \(63 \mathrm{cm}\) from the player's shoulder joint to the point the ball strikes his hand, and his arm remains stiff while it rotates about the shoulder during the catch. The player's hand recoils \(5.00 \mathrm{cm}\) horizontally while he stops the ball. What average torque does the player's arm exert on the ball?

Why do helicopters have two rotors?

Does a particle moving at constant speed in a straight line have angular momentum about a point on the line? About a point not on the line? In either case, is its angular momentum constant?
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