Q: A 30 kg rocket is launched vertically with a thrust of 1500 N and experiences a horizontal wind force of 20 N. What is the rocket's horizontal acceleration?

$$a_x = \frac{F_{\text{wind}}}{m} = \frac{20}{30} = 0.67\ \text{m/s}^2$$

Q: For the rocket in the previous question, what is the vertical acceleration?

$$a_y = \frac{F_{\text{thrust}} - mg}{m} = \frac{1500 - (30 \times 9.8)}{30} = \frac{1500 - 294}{30} = \frac{1206}{30} = 40.2\ \text{m/s}^2$$

Q: A 25 kg cart is spun in a circle of radius 2.0 m with a rope tension of 100 N. What is the speed of the cart?

$$v = \sqrt{\frac{T r}{m}} = \sqrt{\frac{100 \times 2.0}{25}} = \sqrt{8} = 2.83\ \text{m/s}$$

Q: A 1500 kg car turns a corner of radius 50 m on a flat road. If the coefficient of static friction is 1.0, what is the maximum speed before sliding?

$$v_{\text{max}} = \sqrt{\mu_s r g} = \sqrt{1.0 \times 50 \times 9.8} = \sqrt{490} = 22.1\ \text{m/s}$$

Q: A car takes a banked curve of radius 70 m at a 15° angle. What is the speed at which the car can take the curve without friction?

$$v_0$$ = \sqrt{r g \tan \theta} = \sqrt{70 \times 9.8 \times \tan 15^\circ} = \sqrt{70 \times 9.8 \times 0.2679} = \sqrt{183.5} = 13.5\ \text{m/s}$$

Q: A 1.0 kg rock is swung in a horizontal circle on a 1.0 m vine. If the vine breaks at 200 N, what is the maximum angle below horizontal the vine can make before breaking?

$$\sin \theta = \frac{mg}{T} = \frac{1.0 \times 9.8}{200} = 0.049$$; $$\theta = \sin$$^{-1}$$(0.049) = 2.81^\circ$$

Q: A car speeds up around a circle of radius 50 m, starting from rest. The drive wheels supply a steady 525 N force, and the coefficient of static friction is 0.90. How many revolutions before the car slides out of the curve?

$$\Delta \theta_{\text{max}} = \frac{\mu_s m g}{2 F_{\text{wheels}}} \times \frac{1\ \text{rev}}{2\pi\ \text{rad}} = \frac{0.90 \times 1500 \times 9.8}{2 \times 525} \times \frac{1}{2\pi} = 2.0\ \text{rev}$$

Question 1

A 30 kg rocket is launched vertically with a thrust of 1500 N and experiences a horizontal wind force of 20 N. What is the rocket's horizontal acceleration?

Accepted Answer

$$a_x = \frac{F_{	ext{wind}}}{m} = \frac{20}{30} = 0.67\ 	ext{m/s}^2$$

Question 2

For the rocket in the previous question, what is the vertical acceleration?

Accepted Answer

$$a_y = \frac{F_{	ext{thrust}} - mg}{m} = \frac{1500 - (30 	imes 9.8)}{30} = \frac{1500 - 294}{30} = \frac{1206}{30} = 40.2\ 	ext{m/s}^2$$

Question 3

A projectile is launched at three different angles: 30°, 45°, and 60°. Which angle gives the maximum range, assuming no air resistance?

Accepted Answer

45°

Question 4

In uniform circular motion, what is the direction of the acceleration vector?

Accepted Answer

Toward the center of the circle (radial direction)

Question 5

A 25 kg cart is spun in a circle of radius 2.0 m with a rope tension of 100 N. What is the speed of the cart?

Accepted Answer

$$v = \sqrt{\frac{T r}{m}} = \sqrt{\frac{100 	imes 2.0}{25}} = \sqrt{8} = 2.83\ 	ext{m/s}$$

Question 6

A 1500 kg car turns a corner of radius 50 m on a flat road. If the coefficient of static friction is 1.0, what is the maximum speed before sliding?

Accepted Answer

$$v_{	ext{max}} = \sqrt{\mu_s r g} = \sqrt{1.0 	imes 50 	imes 9.8} = \sqrt{490} = 22.1\ 	ext{m/s}$$

Question 7

A car takes a banked curve of radius 70 m at a 15° angle. What is the speed at which the car can take the curve without friction?

Accepted Answer

$$v_0$$ = \sqrt{r g 	an 	heta} = \sqrt{70 	imes 9.8 	imes 	an 15^\circ} = \sqrt{70 	imes 9.8 	imes 0.2679} = \sqrt{183.5} = 13.5\ 	ext{m/s}$$

Question 8

A 1.0 kg rock is swung in a horizontal circle on a 1.0 m vine. If the vine breaks at 200 N, what is the maximum angle below horizontal the vine can make before breaking?

Accepted Answer

$$\sin 	heta = \frac{mg}{T} = \frac{1.0 	imes 9.8}{200} = 0.049$$; $$	heta = \sin$$^{-1}$$(0.049) = 2.81^\circ$$

Question 9

What is the correct free-body diagram for a rock in a sling moving in a horizontal circle?

Accepted Answer

Tension force angled upward and inward, gravity downward

Question 10

A car speeds up around a circle of radius 50 m, starting from rest. The drive wheels supply a steady 525 N force, and the coefficient of static friction is 0.90. How many revolutions before the car slides out of the curve?

Accepted Answer

$$\Delta 	heta_{	ext{max}} = \frac{\mu_s m g}{2 F_{	ext{wheels}}} 	imes \frac{1\ 	ext{rev}}{2\pi\ 	ext{rad}} = \frac{0.90 	imes 1500 	imes 9.8}{2 	imes 525} 	imes \frac{1}{2\pi} = 2.0\ 	ext{rev}$$

Question 11

Which force provides the centripetal acceleration for a car turning a corner on a flat road?

Accepted Answer

Static friction between the tires and the road

Question 12

In uniform circular motion, what is the net force acting on the particle?

Accepted Answer

$$F_{	ext{net}} = \frac{mv^2$}{r}$$, directed toward the center

Question 13

A roller-coaster car goes around a vertical loop. At the top of the loop, what is the minimum speed required for the car to stay on the track?

Accepted Answer

$$v_{	ext{c}} = \sqrt{r g}$$

Question 14

Why does the water stay in the bucket when it is swung in a vertical circle?

Accepted Answer

The net force at the top of the circle is directed toward the center, providing centripetal acceleration

Question 15

What is the effect of Earth's rotation on the measured value of $$g at $$the equator?

Accepted Answer

The measured value of $$g is $$less than it would be on a nonrotating Earth due to the centripetal acceleration

Question 16

In nonuniform circular motion, what causes the particle's speed to change?

Accepted Answer

A tangential force component $$F_{	ext{net}, t}$$

Question 17

A satellite orbits a planet in a circular orbit. What must be true about its acceleration?

Accepted Answer

Its acceleration is always directed toward the center of the planet

Dynamics of Motion in a Plane: Uniform and Nonuniform Circular Motion

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