Essential Physics

Section 1 review

Circular motion can involve rotating, rolling, or orbiting objects. Any object moving in a circle is undergoing centripetal acceleration, which changes the direction of the velocity vector, but not its magnitude (speed). A centripetal force—directed toward the center of the circle—is required to maintain an object in circular motion. Read the text aloud

Read the text aloud

angular velocity, radian (rad), centripetal force, centripetal acceleration

ω = \frac{Δ θ}{Δ t}

v = ω r

a_{c} = \frac{v_{t}^{2}}{r}

F_{c} = \frac{m v_{t}^{2}}{r}

Review problems and questions

Radians and degrees are related by π rad = 180°. Perform the following conversions and draw each one of these angles on a circle.
1. Convert 45° to radians
2. Convert 0.5236 radians to degrees
3. Convert 270° to radians
4. Convert 7.85 radians to degrees
5. Convert 585° to radians

Various angles in degrees and radians

Answer:

0.7854 rad
30°
4.712 rad
450°
10.21 rad

Solution:

$45 ° \times (\frac{π rad}{180 °}) = 0.25 π rad \approx 0.7854 rad$
$0.5236 rad \times (\frac{180 °}{π rad}) = 30 °$
$270 ° \times (\frac{π rad}{180 °}) = 1.5 π rad \approx 4.712 rad$
$7.85 rad \times (\frac{180 °}{π rad}) = 450 °$
$585 ° \times (\frac{π rad}{180 °}) = 3.25 π rad \approx 10.21 rad$

A wheel spins at a rate of 30 revolutions per minute (rpm). What is the angle that a point on the wheel makes in one second? Give your answer in degrees and radians.

Answer: π radians = 180°

Solution:

\begin{array}{l} 30 rev / min \times (\frac{1 min}{60 s}) = 0.5 rev / s \\ 0.5 rev \times (\frac{2 π rad}{1 rev}) = π rad \\ 0.5 rev \times (\frac{360 °}{1 rev}) = 180 ° \end{array}

A race car is moving with a speed of 200 km/hr on a circular section of a race track that has a radius of 300 m. The race car and the driver have a combined mass of 800 kg.
1. What is the magnitude of the centripetal acceleration felt by the driver?
2. What is the centripetal force acting on the car?

Answer:

The driver feels a centripetal acceleration of 10.3 m/s².
The car is under 8,230 N of centripetal force.

Solution:

Asked: centripetal acceleration a_c of the car and driver
Given: race track radius r = 300 m; race car velocity v = 200 km/hr
Relationships: a_c = v²/r, 1,000 m = 1 km, 3,600 s = 1 h
Solve: Convert velocity v from km/hr to m/s: $\begin{array}{l} v = 200 km / hr \times (\frac{1, 000 m}{1 km}) \times (\frac{1 hr}{3, 600 s}) = 55.6 m / s \end{array}$ Solve for a_c: $\begin{array}{l} a_{c} & = & \frac{v^{2}}{r} \\ = & \frac{{(55.6 m / s)}^{2}}{300 m} = 10.3 m / s^{2} \end{array}$ Answer: The driver feels a centripetal acceleration of 10.3 m/s².

Asked: centripetal force F_c on the car
Given: car mass m = 800 kg, centripetal acceleration a_c = 10.3 m/s²
Relationships: F_c = ma_c
Solve: $\begin{array}{l} F_{c} & = & m a_{c} \\ = & (800 kg) \times (10.3 m / s^{2}) = 8, 230 N \end{array}$ Answer: The car is under 8,230 N of centripetal force.

A bicycle moves with a speed of 30 km/hr. If the wheels of the bicycle have a radius of 35 cm, what is the angular speed of the wheels?
Give your answer in rad/s and degrees/s.

Answer: The bicycle has an angular speed of 24 rad/s or 1,400°/s.

Asked: angular speed ω of the wheels
Given: linear speed v = 30 km/hr and radius r = 35 cm = 0.35 m of wheels
Relationships: v = ωr, 1,000 m = 1 km, 3,600 s = 1 hr
Solve: Convert velocity v from km/hr to m/s:

\begin{array}{l} v = 30 km / hr \times (\frac{1, 000 m}{1 km}) \times (\frac{1 hr}{3, 600 s}) = 8.3 m / s \end{array}

Rearrange the equation v = ωr to solve for ω:

\begin{array}{l} ω = \frac{v}{r} = \frac{8.3 m / s}{0.35 m} = 24 rad / s \\ 24 rad / s \times (\frac{180 °}{π rad}) = 1, 400 ° / s \end{array}

Answer: The bicycle has an angular speed of 24 rad/s or 1,400°/s

A 62 kg student rides a Ferris wheel that has a diameter of 50 m and makes one complete rotation every 35 s.
1. What is the angular velocity of the wheel (in radians per second)?
2. What is the linear velocity of the student (in meters per second)?
3. What is the centripetal force acting on the student?


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