Q: A ball on a string is twirled in a horizontal circle of radius $$1.5 m at a $$speed of $$2.7 m/s. $$What angle $$\theta$$ does the string make with the horizontal? (Use $$g = 9.8 m/s$^2)$$

$$\theta = \arctan\left( \dfrac{g r}{v^2$} \right) = 64^\circ$$

Question 1

A ball is spun on a string in a horizontal circle of radius $$r at a $$constant speed $$v. $$What is the correct formula for the period $$T of $$one revolution?

Accepted Answer

$$T = \dfrac{2\pi r}{v}$$

Question 2

Which statement best explains why an object in uniform circular motion is accelerating even though its speed is constant?

Accepted Answer

Its velocity is changing direction, resulting in a nonzero acceleration.

Question 3

What is the direction of the centripetal acceleration for an object in uniform circular motion?

Accepted Answer

Inward, toward the center of the circle

Question 4

A car of mass $$m$$ moves at constant speed $$v$$ around an unbanked curve of radius $$r. If $$the only radial force is friction, what is the maximum speed $$v_{max}$$ the car can have without sliding, given the coefficient of static friction $$\mu_s$$ and gravitational acceleration $$g$$?

Accepted Answer

$$v_{max} = \sqrt{\mu_s g r}$$

Question 5

Which of the following is TRUE regarding the centripetal force in circular motion?

Accepted Answer

It is the vector sum of all radial forces acting toward the center.

Question 6

A ball of mass $$m is $$twirled in a horizontal circle of radius $$r at $$speed $$v. $$What is the tension $$T in $$the string, assuming the weight is negligible?

Accepted Answer

$$T = \dfrac{mv^2$}{r}$$

Question 7

A satellite of mass $$m$$ orbits the Earth in a circular orbit of radius $$r. $$What provides the centripetal force required for its motion?

Accepted Answer

The gravitational force between the Earth and the satellite

Question 8

Which equation correctly relates the gravitational force and the centripetal force for a satellite of mass $$m$$ orbiting Earth (mass $$M) at $$radius $$r$$ with speed $$v$$?

Accepted Answer

$$\dfrac{mv^2$}{r} = \dfrac{GMm}{r^2$}$$

Question 9

A geosynchronous satellite orbits Earth with a period $$T = 24$$ hours. Which formula gives the radius $$r of $$its orbit, given Earth's mass $$M$$ and gravitational constant $$G$$?

Accepted Answer

$$r = \left( \dfrac{$$GMT^2$$}{4\pi^2$} \$$right)^{1/3}$$

Question 10

Why do astronauts in the International Space Station appear to be 'weightless'?

Accepted Answer

They are in continuous free fall, accelerating at the same rate as the station.

Question 11

A ball on a string is swung in a vertical circle. At the top of the circle, what is the minimum speed $$v_{min}$$ required to keep the string taut, given radius $$r$$ and gravitational acceleration $$g$$?

Accepted Answer

$$v_{min} = \sqrt{g r}$$

Question 12

A ball of mass $$m is $$twirled in a circle of radius $$r at $$speed $$v. $$What is the formula for its centripetal acceleration $$a_c$$?

Accepted Answer

$$a_c = \dfrac{v^2$}{r}$$

Question 13

A ball on a string is twirled in a horizontal circle of radius $$1.5 m at a $$speed of $$2.7 m/s. $$What angle $$	heta$$ does the string make with the horizontal? (Use $$g = 9.8 m/s$^2)$$

Accepted Answer

$$	heta = \arctan\left( \dfrac{g r}{v^2$} ight) = 64^\circ$$

Question 14

Which of the following best describes the 'centrifugal force' experienced in a rotating reference frame?

Accepted Answer

It is a fictitious force that appears due to non-inertial reference frames.

Question 15

A passenger in NASA's 'Vomit Comet' experiences weightlessness at the top of a parabolic trajectory. What is the condition for this sensation?

Accepted Answer

The normal force from the seat is zero ($$F_N = 0)$$

Question 16

If a space station is designed as a rotating ring to create 'artificial gravity', what must be true for the sensation to match Earth's gravity ($$g$$)?

Accepted Answer

The centripetal acceleration at the ring's edge must equal $$g$$

Question 17

Which principle allows artificial gravity to be created in a rotating space station?

Accepted Answer

Einstein's Principle of Equivalence

Question 18

A ball moves in a circle under the influence of three radial forces: $$F_1$$ and $$F_2$$ point inward, $$F_3$$ points outward. What is the net centripetal force?

Accepted Answer

$$F_c = F_1$ + F_2$ - F_3$

Uniform Circular Motion and Centripetal Forces: Study Notes

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