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What is the ideal speed to take a \(100{\rm{ m}}\) radius curve banked at \(20.0^\circ \) angle?

Short Answer

Expert verified

The ideal speed is \(18.9{\rm{ m}}/{\rm{s}}\).

Step by step solution

01

Definition of Banking of road

To supply the centre gravity force required for a vehicle to perform a safe turn, the outside edge of curved roadways is raised above the inner edge. This is referred to as road banking.

02

Calculating Ideal speed 

The ideal speed for a banked road is,

\(v = \sqrt {Rg\tan \theta } \)

Here, \(R\) is the radius of the turn, \(g\) is the acceleration due to gravity, and \(\theta \) is the ideal banking angle.

Substitute \(100{\rm{ m}}\) for \(R\), \(9.8{\rm{ m}}/{{\rm{s}}^2}\) for \(g\), and \(20.0^\circ \) for \(\theta \),

\(\begin{array}{c}v = \sqrt {\left( {100{\rm{ m}}} \right) \times \left( {9.8{\rm{ m}}/{{\rm{s}}^2}} \right) \times \tan \left( {20.0^\circ } \right)} \\ = 18.9{\rm{ m}}/{\rm{s}}\end{array}\)

Hence, the ideal speed is \(18.9{\rm{ m}}/{\rm{s}}\).

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Most popular questions from this chapter

A mother pushes her child on a swing so that his speed is9.00m/s at the lowest point of this path. The swing is suspended 2.00 mabove the childโ€™s center of mass.

(a) What is the magnitude of the centripetal acceleration of the child at the low point?

(b) What is the magnitude of the force the child exerts on the seat if his mass is 18.0 kg?

(c) What is unreasonable about these results?

(d) Which premises are unreasonable or inconsistent?

A large centrifuge, like the one shown in Figure (a), is used to expose aspiring astronauts to accelerations similar to those experienced in rocket launches and atmospheric re-entries. (a) At what angular velocity is the centripetal acceleration \(10g\) if the rider is \(15.0{\rm{ m}}\) from the centre of rotation? (b) The riderโ€™s cage hangs on a pivot at the end of the arm, allowing it to swing outward during rotation as shown in Figure (b). At what angle \(\theta \) below the horizontal will the cage hang when the centripetal acceleration is \(10g\)? (Hint: The arm supplies centripetal force and supports the weight of the cage. Draw a free body diagram of the forces to see what the angle \(\theta \) should be.)



*Figure (a) NASA centrifuge used to subject trainees to accelerations similar to those experienced in rocket launches and re-entries. (Credit: NASA) (b) Rider in cage showing how the cage pivots outward during rotation. This allows the total force exerted on the rider by the cage to be along its axis at all times.

In one amusement park ride, riders enter a large vertical barrel and stand against the wall on its horizontal floor. The barrel is spun up and the floor drops away. Riders feel as if they are pinned to the wall by a force something like the gravitational force. This is a fictitious force sensed and used by the riders to explain events in the rotating frame of reference of the barrel. Explain in an inertial frame of reference (Earth is nearly one) what pins the riders to the wall, and identify all of the real forces acting on them.

What is the direction of the force exerted by the car on the passenger as the car goes over the top of the amusement ride under the following circumstances: (a) the car goes over the top at such a speed that the gravitational force is the only force acting? (b) The car goes over the top faster than this speed? (c) The car goes over the top slower than this speed?

Semi-trailer trucks have an odometer on one hub of a trailer wheel. The hub is weighted so that it does not rotate, but it contains gears to count the number of wheel revolutionsโ€”it then calculates the distance travelled. If the wheel has a 1.15 m diameter and goes through 200,000 rotations, how many kilometres should the odometer read?

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