Question

1.A student is riding their bmx bike on a track that forms a vertical circular loop with a diameter of 6.0m. In order to reach the top of the loop and not fall, the student would have to be travelling at a speed of ___m/s. (Give your answer with the correct number of significant digits and do no include units).

2.A student has a weight of 650 N. While riding on a roller-coaster this same student has an apparent weight of 1.97 x 10³ N at the bottom of a loop that has a radius of 17.0 m. The speed of the roller-coaster is ____m/s. (Give your answer with 3 sig digs and do not include units).

3.A string requires a 191.0 N force in order to break. A 1.75 kg mass is tied to this string and whirled in a vertical circle with a radius of 1.92 m. The maximum speed that this mass can be whirled without breaking the string is ___m/s. (Give your answer with 3 sig digs, and do not include units)

Answer 1

1)

$$N+mg=\dfrac{mv^2}{r}.$$

In order to reach the top of the loop and not fall $N$ must be 0. Therefore, we get:

$$v=\sqrt{gr}=\sqrt{9.8\ \dfrac{m}{s^2}\cdot3.0\ m}=5.42\ \dfrac{m}{s}.$$

2)

$$F_c=N-mg,$$ $$\dfrac{mv^2}{r}=N-mg,$$ $$v=\sqrt{\dfrac{Nr}{m}-gr},$$ $$v=\sqrt{\dfrac{1.97\cdot10^3\ N\cdot17.0\ m}{\dfrac{650\ N}{9.8\ \dfrac{m}{s^2}}}-9.8\ \dfrac{m}{s^2}\cdot17.0\ m}=18.4\ \dfrac{m}{s}.$$

3)

$$T-F_g=F_c,$$ $$T-F_g=\dfrac{mv^2}{r},$$ $$v=\sqrt{\dfrac{(T-mg)r}{m}},$$ $$v=\sqrt{\dfrac{(191\ N-1.75\ kg\cdot9.8\ \dfrac{m}{s^2})\cdot1.92\ m}{1.75\ kg}}=13.81\ \dfrac{m}{s}.$$