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Chapter 7: Problem 23

A certain light truck can go around a flat curve having a radius of \(150 \mathrm{~m}\) with a maximum speed of \(32.0 \mathrm{~m} / \mathrm{s}\). With what maximum speed can it go around a curve having a radius of \(75.0 \mathrm{~m}\) ?

Short Answer

Expert verified
The maximum speed for the light truck to go around a curve having a radius of 75.0 m is approximately 22.6 m/s.

Step by step solution

01

Identify given values and unknowns

The given values are the radius of the first curve (150 m) and the speed of the truck around that curve (32.0 m/s). The speed of the truck around the second curve (radius = 75.0 m) is unknown.
02

Formulate our proportion

We know that the centripetal force for a curve is given by \( F_c = m v^2 / r \). Since the mass of the truck is the same in both cases, the forces are proportional to each other: \( F_{c1} / F_{c2} = v_{1}^2 / v_{2}^2 = r_1 / r_2 \)
03

Substitute the known values

From the problem, we know that \( r_1 = 150 m\), \( v_1 = 32.0 m/s\), and \( r_2 = 75.0 m\). When we plug these values into our equation from step 2, we have: \( (32.0 m/s)^2 / v_{2}^2 = 150 m / 75.0 m \).
04

Solve for the unknown

To solve for \( v_2\), rearrange the equation as: \( v_{2} = \sqrt{ (32.0 m/s)^2 * 75.0 m / 150 m } \). After performing the calculation, the maximum speed for the truck on the second curve is approximately 22.6 m/s. So, for the curve with a radius of 75 m, the truck should not exceed this speed.

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Most popular questions from this chapter

The pilot of an airplane executes a constantspeed loop-the-loop maneuver in a vertical circle as in Figure \(7.15 \mathrm{~b}\). The speed of the airplane is \(2.00 \times\) \(10^{2} \mathrm{~m} / \mathrm{s}\), and the radius of the circle is \(3.20 \times 10^{3} \mathrm{~m}\). (a) What is the pilot's apparent weight at the lowest point of the circle if his true weight is \(712 \mathrm{~N}\) ? (b) What is his apparent weight at the highest point of the circle? (c) Describe how the pilot could experience weightlessness if both the radius and the speed can be varied. Note: His apparent weight is equal to the magnitude of the force exerted by the seat on his body. Under what conditions does this occur? (d) What speed would have resulted in the pilot experiencing weightlessness at the top of the loop?

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A \(45.0\)-cm diameter disk rotates with a constant angular acceleration of \(2.50 \mathrm{rad} / \mathrm{s}^{2}\). It starts from rest at \(t=\) 0 , and a line drawn from the center of the disk to a point \(P\) on the rim of the disk makes an angle of \(57.3^{\circ}\) with the positive \(x\)-axis at this time. At \(t=2.30 \mathrm{~s}\), find (a) the angular speed of the wheel, (b) the linear velocity and tangential acceleration of \(P\), and (c) the position of \(P\) (in degrees, with respect to the positive \(x\)-axis).

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