Q: A 6.0 kg block slides down a 35° incline with a coefficient of kinetic friction $$\mu_k = 0.25. $$What is the normal force acting on the block?

$$F_N = mg \cos(35^\circ) = 6.0 \times 9.8 \times \cos(35^\circ) = 48.2\ \mathrm{N}$$

Q: A car of mass 1000 kg rounds a curve of radius 80 m at 20 m/s. What is the required centripetal force?

$$F_c = m \frac{v^2$}{r} = 1000 \times \frac{20^2$}{80} = 5000\ \mathrm{N}$$

Q: If a projectile is launched at a speed $$v_0$$ and angle $$\theta$$, which launch angle produces the maximum range (neglecting air resistance)?

$$45^\circ$$, because $$\sin(2\theta) is $$maximized at $$\theta = 45^\circ$$

Q: A 2.0 kg block slides down a frictionless 4.0 m ramp at 30° and enters a vertical circular loop of radius 1.5 m. What is the minimum starting height $$h$$ needed for the block to just complete the loop?

$$h = \frac{5}{2}R = 2.5 \times 1.5 = 3.75\ \mathrm{m}$$

Q: A projectile lands 50 m away when launched at 45°. What was its initial speed? (Assume $$g = 9.8\ \mathrm{m/s^2$})$$

$$v_0$$ = \sqrt{\frac{R g}{\sin(2\theta)}} = \sqrt{\frac{50 \times 9.8}{\sin(90^\circ)}} = \sqrt{490} \approx 22.1\ \mathrm{m/s}$$

Q: A projectile is launched at 40 m/s at an angle $$\theta. At$$ $$t = 2 s$$, its velocity vector is (30, 5) m/s. What was the launch angle?

$$\theta = \arctan\left(\frac{v_{0y}}{v_{0x}}\right) = \arctan\left(\frac{25.6}{32}\right) \approx 39.5^\circ$$

Question 1

A vector has a magnitude of 10 N and points at 30° above the +x axis. Which of the following correctly describes the relationship between its x- and y-components?

Accepted Answer

The x-component is larger than the y-component because $$\cos(30^\circ) \u003e \sin(30^\circ).$$

Question 2

A 6.0 kg block slides down a 35° incline with a coefficient of kinetic friction $$\mu_k = 0.25. $$What is the normal force acting on the block?

Accepted Answer

$$F_N = mg \cos(35^\circ) = 6.0 	imes 9.8 	imes \cos(35^\circ) = 48.2\ \mathrm{N}$$

Question 3

A projectile is launched horizontally from a 20 m high cliff. What determines the time the projectile spends in the air?

Accepted Answer

The vertical height of the cliff and gravity, using $$t = \sqrt{\frac{2h}{g}}$$

Question 4

A car of mass 1000 kg rounds a curve of radius 80 m at 20 m/s. What is the required centripetal force?

Accepted Answer

$$F_c = m \frac{v^2$}{r} = 1000 	imes \frac{20^2$}{80} = 5000\ \mathrm{N}$$

Question 5

If a projectile is launched at a speed $$v_0$$ and angle $$	heta$$, which launch angle produces the maximum range (neglecting air resistance)?

Accepted Answer

$$45^\circ$$, because $$\sin(2	heta) is $$maximized at $$	heta = 45^\circ$$

Question 6

A 2.0 kg block slides down a frictionless 4.0 m ramp at 30° and enters a vertical circular loop of radius 1.5 m. What is the minimum starting height $$h$$ needed for the block to just complete the loop?

Accepted Answer

$$h = \frac{5}{2}R = 2.5 	imes 1.5 = 3.75\ \mathrm{m}$$

Question 7

A projectile lands 50 m away when launched at 45°. What was its initial speed? (Assume $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$v_0$$ = \sqrt{\frac{R g}{\sin(2	heta)}} = \sqrt{\frac{50 	imes 9.8}{\sin(90^\circ)}} = \sqrt{490} \approx 22.1\ \mathrm{m/s}$$

Question 8

A piston of mass 0.6 kg moves in a circle of radius 0.04 m. The maximum allowable force on the rod is 15,000 N. What is the maximum angular velocity $$\omega (in $$rad/s) the piston can have?

Accepted Answer

$$\omega = \sqrt{\frac{F_{max}}{m r}} = \sqrt{\frac{15000}{0.6 	imes 0.04}} = \sqrt{625000} = 790.6\ \mathrm{rad/s}$$

Question 9

A block slides down a frictionless ramp. Why does its acceleration not depend on its mass?

Accepted Answer

Because both the gravitational force and inertia are proportional to mass, so $$a = g \sin	heta is $$independent of mass.

Question 10

A projectile is launched at 40 m/s at an angle $$	heta. At$$ $$t = 2 s$$, its velocity vector is (30, 5) m/s. What was the launch angle?

Accepted Answer

$$	heta = \arctan\left(\frac{v_{0y}}{v_{0x}}ight) = \arctan\left(\frac{25.6}{32}ight) \approx 39.5^\circ$$

Question 11

If the net force on an object is zero, which of the following must be true?

Accepted Answer

The object moves at constant velocity or remains at rest.

Question 12

A 6.0 kg block is pulled upward at a 25° angle above the horizontal with a rope, while sliding to the right at constant velocity on a table with kinetic friction. Which of the following is true about the forces acting on the block?

Accepted Answer

The net force in both x and y directions is zero.

Question 13

If the speed of an object in uniform circular motion doubles, what happens to its centripetal acceleration?

Accepted Answer

It increases by a factor of 4, since $$a_c = \frac{v^2$}{r}.$$

Question 14

A ball is launched horizontally at 15 m/s from a height of 5 m (Ball A) and another from 20 m (Ball B). Which ball lands farther from the base of the cliff?

Accepted Answer

Ball B, because it spends more time in the air and has the same horizontal speed.

Question 15

Why is 'centrifugal force' not considered a real force in an inertial frame of reference?

Accepted Answer

Because it is a fictitious force that appears only in rotating (non-inertial) frames.

Question 16

A 25 kg crate is acted on by two forces: $$F_1 = 120$$\ \mathrm{N}$$ at 40° above +x and F_2$ = 200\ \mathrm{N} at 25$$° below +x. What is the net force in the x-direction?

Accepted Answer

$$F_{net,x} = 120 \cos(40^\circ) + 200 \cos(25^\circ) = 91.98 + 181.32 = 273.3\ \mathrm{N}$$

Question 17

On an inclined plane, what component of gravity causes the object to accelerate down the ramp?

Accepted Answer

$$mg \sin	heta$$

Question 18

A road is designed for cars traveling at 30 m/s with a maximum safe sideways acceleration of $$0.1g. $$What is the minimum safe radius of curvature for the road?

Accepted Answer

$$r = \frac{v^2$}{a_{max}} = \frac{30^2$}{0.1 	imes 9.8} = \frac{900}{0.98} \approx 918\ \mathrm{m}$$

Study Guide: Motion in 2D, Projectiles, Inclined Planes, and Circular Motion — Step-by-Step Guidance

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