Ten sense-making tasks on motion in a circle. Explain your reasoning for every answer. The explanation counts as much as the answer.
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Six cars drive around flat (unbanked) circular curves at constant speed. Friction from the road is the only horizontal force on each car.
| Car | Mass | Speed | Radius of curve |
|---|---|---|---|
| A | 1000 kg | 10 m/s | 50 m |
| B | 1500 kg | 20 m/s | 100 m |
| C | 800 kg | 20 m/s | 200 m |
| D | 2000 kg | 10 m/s | 25 m |
| E | 1200 kg | 30 m/s | 100 m |
| F | 500 kg | 5 m/s | 25 m |
Rank the cars by the minimum coefficient of static friction needed to make it around the curve without skidding, from greatest to least. If any are equal, show the tie. If the ranking cannot be determined, say so.
Greatest 1 ____ 2 ____ 3 ____ 4 ____ 5 ____ 6 ____ Least
Explain your reasoning.
Back to topA ball on a string is whirled in a horizontal circle (top view below), moving counterclockwise. At the instant shown, the string breaks. Three students predict the ball's path, as seen from above.
Ava: "Path 1. The ball was being pulled outward the whole time; the string was holding it in. Once the string breaks, it flies straight out, away from the center."
Ben: "Path 2. The ball has been going in a circle, so it keeps curving for a while and then gradually straightens out."
Cam: "Path 3. Once the string breaks, nothing pushes the ball sideways, so it keeps going in a straight line in the direction it was moving at that instant."
Which student(s), if any, do you agree with? Explain what is right or wrong about each statement.
Back to topA car rounds an icy (frictionless) banked curve at exactly the right speed to make the turn. The view is from behind the car; the center of the curve is to the left. A student draws this free-body diagram and labels the horizontal arrow $F_c$, "the centripetal force."
What, if anything, is wrong with this free-body diagram? If something is wrong, explain the error and describe the correct diagram. If nothing is wrong, explain why it is correct.
Back to topA rider of mass $m$ rides a roller coaster. V is the bottom of a valley, H is the top of a hill, T is the top of a vertical loop (rider upside down), and B is the bottom of the loop. For each situation, choose how the normal force from the seat on the rider compares to $mg$:
A. greater than $mg$ B. equal to $mg$ C. less than $mg$ D. cannot be determined
| Situation | Choice |
|---|---|
| (a) Moving through V | |
| (b) Moving over H (still on the track) | |
| (c) On the flat section at the far right, constant speed | |
| (d) Moving through T (speed not given) | |
| (e) Moving through T at the minimum speed needed to stay on the track | |
| (f) Moving through B | |
| (g) Stopped at rest on top of H (stalled) |
Explain the rule you used. What do the "greater than" situations have in common?
Back to topTwo identical coins sit on a turntable that spins at a constant rate. Coin 1 is a distance $r$ from the center and Coin 2 is $2r$ from the center. Neither coin slides.
For each quantity, state whether it is greater for Coin 1, greater for Coin 2, the same for both, or cannot be determined. If it's different, give the ratio.
| Quantity | Answer |
|---|---|
| (a) Time for one revolution (period) | |
| (b) Speed | |
| (c) Centripetal acceleration | |
| (d) Friction force on the coin |
Explain each answer. If the turntable is slowly sped up, which coin starts to slide first? Why?
Back to topA 0.50 kg ball on a light string moves in a horizontal circle at constant speed, with the string making 37° with the vertical. Use $\sin37^\circ = 0.60$, $\cos37^\circ = 0.80$.
The equation below is Newton's 2nd law applied to one object at one instant:
$$F_N - (60\ \text{kg})(10\ \text{m/s}^2) = (60\ \text{kg})\,\frac{(12\ \text{m/s})^2}{8.0\ \text{m}}$$
A car drives counterclockwise around a circular track at constant speed (top view).
A curve of radius 120 m is banked at 37°. Find the speed at which a car can round the curve with no friction at all. Use $g = 10\ \text{m/s}^2$, $\sin37^\circ = 0.60$, $\cos37^\circ = 0.80$.
Student solution:
Perpendicular to the road: $F_N = mg\cos37^\circ$
Toward the center: $F_N\sin37^\circ = \dfrac{mv^2}{r}$
$mg\cos37^\circ\sin37^\circ = \dfrac{mv^2}{r} \Rightarrow v^2 = gr(0.80)(0.60) = (10\ \text{m/s}^2)(120\ \text{m})(0.48)$
$v = 24\ \text{m/s}$
Identify the error in the solution. Explain the physics of why it's an error, then find the correct speed.
Back to topA 1000 kg car drives over the crest of a hill. The top of the hill is part of a circle of radius 40 m (the drawing is not to scale). The car's speed at the top is 25 m/s.
Jordan writes: $mg - F_N = \dfrac{mv^2}{r} \Rightarrow F_N = (1000\ \text{kg})(10\ \text{m/s}^2) - \dfrac{(1000\ \text{kg})(25\ \text{m/s})^2}{40\ \text{m}} = -5625\ \text{N}$. "So the road pulls the car down with 5625 N."