Q: A tetherball swings in a horizontal circle around a pole, attached by a string. If the mass of the ball is $$m$$, the length of the string is $$L$$, and the ball moves at a constant speed $$v$$, what is the angle $$\theta$$ that the string makes with the vertical? (Neglect air resistance.)

$$\theta = \arctan\left(\frac{v^2$}{gL}\right)$$

Question 1

A tetherball swings in a horizontal circle around a pole, attached by a string. If the mass of the ball is $$m$$, the length of the string is $$L$$, and the ball moves at a constant speed $$v$$, what is the angle $$	heta$$ that the string makes with the vertical? (Neglect air resistance.)

Accepted Answer

$$	heta = \arctan\left(\frac{v^2$}{gL}ight)$$

Question 2

A motorboat can travel at a speed of $$5\ \mathrm{m/s} in $$still water. If it tries to cross a river flowing east at $$3\ \mathrm{m/s}$$, what is the boat's velocity relative to the ground if it heads due north?

Accepted Answer

$$\sqrt{5^2$ + 3^2$} = 5.83\ \mathrm{m/s} at$$ $$\arctan(3/5) = 31^\circ$$ east of north

Question 3

If a tetherball is moving in a horizontal circle and the string makes an angle $$	heta$$ with the vertical, which force provides the necessary centripetal force for the ball's circular motion?

Accepted Answer

The horizontal component of the tension in the string

Question 4

A boat heads directly across a $$200\ \mathrm{m}$$ wide river with a speed of $$4\ \mathrm{m/s}$$ relative to the water. The river flows at $$3\ \mathrm{m/s}. $$How far downstream will the boat land on the opposite bank?

Accepted Answer

$$150\ \mathrm{m}$$ downstream

Question 5

A tetherball of mass $$m$$ swings in a circle of radius $$r at $$constant speed $$v. $$What is the tension $$T in $$the string?

Accepted Answer

$$T = \sqrt{(mg)^2$ + (m\frac{v^2$}{r})^2$}$$

Question 6

A motorboat moves at $$6\ \mathrm{m/s} in $$still water. If it wants to go straight across a river flowing at $$8\ \mathrm{m/s}$$, at what angle upstream should it head relative to the current?

Accepted Answer

$$\arcsin\left(\frac{8}{6}\right) is $$not possible, so the boat cannot make it directly across

Question 7

If a tetherball is moving faster, what happens to the angle $$	heta$$ the string makes with the vertical?

Accepted Answer

The angle increases

Question 8

A boat travels at $$10\ \mathrm{m/s} in $$still water. The river flows at $$6\ \mathrm{m/s}. $$What is the minimum time to cross a $$120\ \mathrm{m}$$ wide river?

Accepted Answer

$$12\ \mathrm{s}$$

Question 9

Which of the following best describes the direction of the net force acting on a tetherball moving in a horizontal circle?

Accepted Answer

Radially inward toward the pole

Question 10

A boat heads $$30^\circ$$ upstream relative to the current, with a speed of $$8\ \mathrm{m/s}$$ relative to the water. The river flows at $$4\ \mathrm{m/s}. $$What is the boat's speed relative to the ground?

Accepted Answer

$$\sqrt{(8\sin 30^\$$circ)^2$$ + (8\cos 30^\circ - 4)^2$}$$

Question 11

If the string of a tetherball breaks while the ball is moving in a circle, what path will the ball follow immediately after?

Accepted Answer

It will move in a straight line tangent to the circle at the point of release

Question 12

A boat can move at $$7\ \mathrm{m/s} in $$still water. If it wants to cross a $$210\ \mathrm{m}$$ wide river flowing at $$5\ \mathrm{m/s}$$, what is the resultant velocity of the boat relative to the ground?

Accepted Answer

$$\sqrt{7^2$ + 5^2$} = 8.6\ \mathrm{m/s}$$

Question 13

Which of the following statements about relative velocity is correct?

Accepted Answer

Relative velocity depends on the observer's frame of reference

Question 14

A tetherball of mass $$0.5\ \mathrm{kg}$$ swings in a circle of radius $$1.2\ \mathrm{m} at a $$speed of $$3\ \mathrm{m/s}. $$What is the magnitude of the horizontal component of the tension in the string?

Accepted Answer

$$0.5 	imes \frac{3^2$}{1.2} = 3.75\ \mathrm{N}$$

Question 15

A boat travels at $$5\ \mathrm{m/s} in $$still water. The river flows at $$4\ \mathrm{m/s}. If $$the boat heads directly across, what is the angle downstream from straight across at which it lands?

Accepted Answer

$$\arctan\left(\frac{4}{5}\right) = 38.7^\circ$$

Physics Study Notes: Tetherball and Relative Velocity in River Motion

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