Q: A car is driving on a banked curve of radius $$r$$ and angle $$\theta. $$What is the optimal speed $$v_0$$ for the car to take the curve without relying on friction?

$$v_0$$ = \sqrt{r g \tan \theta}$$

Question 1

A 1500 kg car is driving around a flat, unbanked curve of radius 50 m. If the coefficient of static friction between the tires and the road is 0.80, what is the maximum speed the car can have without sliding?

Accepted Answer

$$v_{max} = \sqrt{\mu_s g r}$$

Question 2

A rollercoaster car of mass $$m$$ goes around a loop of radius $$r. $$What is the minimum speed at the top of the loop for the car to stay on the track?

Accepted Answer

$$v_{min} = \sqrt{r g}$$

Question 3

A satellite is in a circular orbit near the surface of a planet of radius $$r$$ and gravitational acceleration $$g. $$What is the orbital speed?

Accepted Answer

$$v_{orbit} = \sqrt{r g}$$

Question 4

A car is driving on a banked curve of radius $$r$$ and angle $$	heta. $$What is the optimal speed $$v_0$$ for the car to take the curve without relying on friction?

Accepted Answer

$$v_0$$ = \sqrt{r g 	an 	heta}$$

Question 5

In uniform circular motion, which direction does the centripetal acceleration vector point?

Accepted Answer

Toward the center of the circle

Question 6

A child of mass 20 kg sits on a cart of mass 5 kg, attached to a 2 m rope. If the tension in the rope is 100 N, what is the angular speed $$\omega of $$the cart?

Accepted Answer

$$\omega = \sqrt{\frac{T}{m_{total} r}}$$

Question 7

Which of the following statements about centrifugal force is correct?

Accepted Answer

It is a fictitious force observed in a non-inertial reference frame.

Question 8

A particle moves in a circle of radius $$r$$ with constant speed $$v. $$What is the magnitude of the net force acting on it?

Accepted Answer

$$F_{net} = \frac{m v^2$}{r}$$

Question 9

A car takes a banked curve at a speed less than the optimal speed $$v_0. $$What role does friction play?

Accepted Answer

Friction points uphill to prevent the car from slipping down the bank.

Question 10

In a loop-the-loop, what happens if the car's speed at the top is less than the critical speed $$v_C$$?

Accepted Answer

The car loses contact with the track and follows a parabolic trajectory.

Question 11

For a satellite in low Earth orbit, which of the following best explains why astronauts experience weightlessness?

Accepted Answer

They are in free fall, constantly accelerating toward Earth but never reaching the surface.

Question 12

A particle moves in a circle with both radial and tangential forces acting. Which equation correctly describes the tangential acceleration?

Accepted Answer

$$a_t = \frac{d v_t}{d t}$$

Question 13

Which coordinate system is most convenient for analyzing uniform circular motion?

Accepted Answer

The $$r t z$$ coordinate system, where $$r is $$radial, $$t is $$tangential, and $$z is $$perpendicular to the plane.

Question 14

A car drives around a flat, 50-m-diameter track, starting from rest. The drive wheels supply a steady 500 N force. If the coefficient of static friction is 0.80, what determines when the car slides out of the curve?

Accepted Answer

When the required centripetal force exceeds the maximum static friction force.

Question 15

In Newtonian mechanics, what is the first step in the problem-solving strategy?

Accepted Answer

Model the object as a particle and make simplifying assumptions.

Question 16

Which force is responsible for keeping a ball moving in a circle when swung on a string?

Accepted Answer

Tension in the string acts as the centripetal force.

Question 17

If a car takes a banked curve at a speed greater than $$v_0$$, what direction does friction act?

Accepted Answer

Friction points downhill to prevent the car from slipping up the bank.

Question 18

What is the period $$T of a $$satellite in a circular orbit near the surface of a planet?

Accepted Answer

$$T = 2 \pi \sqrt{\frac{r}{g}}$$

Question 19

In uniform circular motion, what is the value of tangential acceleration $$a_t$$?

Accepted Answer

$$a_t = 0$$

Dynamics in Two Dimensions: Circular Motion and Applications

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