Q: A 3.00-kg mass moves in a plane with its $$x$$ and $$y$$ coordinates given by $$x = 5t^2$ - 1$$ and $$y = 3t^3$ + 2$$, where $$x$$ and $$y$$ are in meters and $$t is in $$seconds. What is the magnitude of the net force acting on the mass at $$t = 2.00 s$$?

$$F = 3.00 \times \sqrt{20^2$ + 108^2$} = 3.00 \times 109.85 = 329.55\ \mathrm{N}$$

Q: The distance between two telephone poles is $$5.00\ \mathrm{m}. $$When a $$1.00\ \mathrm{kg}$$ bird lands on the wire midway between the poles, the wire sags $$0.020\ \mathrm{m}. $$What is the tension in the wire at the lowest point?

$$T = \frac{mg \cdot (L/2)}{y} = \frac{1.00 \times 9.8 \times 2.5}{0.020} = 1225\ \mathrm{N}$$

Q: A mass $$m is $$suspended from a string of length $$L in a $$car accelerating horizontally with acceleration $$a. $$What is the angle $$\theta$$ the string makes with the vertical?

$$\theta = \tan$$^{-1}$$\left(\frac{a}{g}\right)$$

Q: A cart with a suspended mass is accelerated so that the string makes a constant angle $$\theta$$ with the vertical. If $$\theta = 30^\circ$$, what is the acceleration of the cart?

$$a = g \tan 30^\circ = 9.8 \times 0.577 = 5.66\ \mathrm{m/s^2$}$$

Q: If two people pull on a $$209\ \mathrm{kg}$$ boat in opposite directions and the acceleration is $$0.518\ \mathrm{m/s^2$} to $$the right, what is the net force acting on the boat?

$$F_{net} = 209 \times 0.518 = 108.2\ \mathrm{N}$$

Q: A force $$F is $$applied to two masses $$m_1$$ and $$m_2$$ connected by a string on a frictionless surface. If $$m_1 = 2$$\ \mathrm{kg}$$, m_2$ = 3\ \mathrm{kg}$$, and $$F = 10\ \mathrm{N}$$, what is the tension in the string?

$$T = \frac{m_1$ F}{m_1$ + m_2$} = \frac{2 \times 10}{5} = 4\ \mathrm{N}$$

Question 1

A 3.00-kg mass moves in a plane with its $$x$$ and $$y$$ coordinates given by $$x = 5t^2$ - 1$$ and $$y = 3t^3$ + 2$$, where $$x$$ and $$y$$ are in meters and $$t is in $$seconds. What is the magnitude of the net force acting on the mass at $$t = 2.00 s$$?

Accepted Answer

$$F = 3.00 	imes \sqrt{20^2$ + 108^2$} = 3.00 	imes 109.85 = 329.55\ \mathrm{N}$$

Question 2

The distance between two telephone poles is $$5.00\ \mathrm{m}. $$When a $$1.00\ \mathrm{kg}$$ bird lands on the wire midway between the poles, the wire sags $$0.020\ \mathrm{m}. $$What is the tension in the wire at the lowest point?

Accepted Answer

$$T = \frac{mg \cdot (L/2)}{y} = \frac{1.00 	imes 9.8 	imes 2.5}{0.020} = 1225\ \mathrm{N}$$

Question 3

In the spring scale system shown (with two $$10.0\ \mathrm{kg}$$ masses and a frictionless pulley), what does each spring scale read in newtons?

Accepted Answer

$$98\ \mathrm{N}$$

Question 4

A simple accelerometer is constructed by suspending a mass from a string of length $$L$$ tied to a cart. If the cart accelerates with acceleration $$a at an $$angle $$	heta$$ with the vertical, what is the expression for $$a in $$terms of $$g$$ and $$	heta$$?

Accepted Answer

$$a = g 	an 	heta$$

Question 5

Two masses $$m_1$$ and $$m_2$$ are connected by a light string on a frictionless horizontal surface. A force $$F is $$applied to $$m_2 ($$see figure). What is the acceleration $$a of $$the system?

Accepted Answer

$$a = \frac{F}{m_1$ + m_2$}$$

Question 6

In the same two-mass system, what is the tension $$T in $$the string connecting $$m_1$$ and $$m_2 if $$the force $$F is $$applied to $$m_2$$?

Accepted Answer

$$T = \frac{m_1$ F}{m_1$ + m_2$}$$

Question 7

Two people pull as hard as they can on ropes attached to a $$209\ \mathrm{kg}$$ boat. If they pull in the same direction, the boat has an acceleration of $$1.52\ \mathrm{m/s^2$}. If $$they pull in opposite directions, the boat has an acceleration of $$0.518\ \mathrm{m/s^2$} to $$the right. What is the magnitude of the larger force?

Accepted Answer

$$F_1$$ = \frac{($$a_1$$ + $$a_2)m$$}{2} = \frac{(1.52 + 0.518) 	imes 209}{2} = 212\ \mathrm{N}$$

Question 8

A mass $$m is $$suspended from a string of length $$L in a $$car accelerating horizontally with acceleration $$a. $$What is the angle $$	heta$$ the string makes with the vertical?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{a}{g}ight)$$

Question 9

A $$1.00\ \mathrm{kg}$$ bird lands at the midpoint of a $$5.00\ \mathrm{m}$$ wire, causing it to sag $$0.020\ \mathrm{m}. $$What is the vertical component of the tension in the wire at the midpoint?

Accepted Answer

$$T_y = mg = 9.8\ \mathrm{N}$$

Question 10

A $$3.00\ \mathrm{kg}$$ mass moves in a plane with $$x = 5t^2$ - 1$$ and $$y = 3t^3$ + 2. $$What is the $$y-$$component of the acceleration at $$t = 2.00 s$$?

Accepted Answer

$$a_y = 36\ \mathrm{m/s^2$}$$

Question 11

A cart with a suspended mass is accelerated so that the string makes a constant angle $$	heta$$ with the vertical. If $$	heta = 30^\circ$$, what is the acceleration of the cart?

Accepted Answer

$$a = g 	an 30^\circ = 9.8 	imes 0.577 = 5.66\ \mathrm{m/s^2$}$$

Question 12

If the spring scales in the equilibrium system (with two $$10.0\ \mathrm{kg}$$ masses) are replaced with $$20.0\ \mathrm{kg}$$ masses, what will the new reading be on each scale?

Accepted Answer

$$196\ \mathrm{N}$$

Question 13

A $$3.00\ \mathrm{kg}$$ mass moves in a plane with $$x = 5t^2$ - 1$$ and $$y = 3t^3$ + 2. $$What is the $$x-$$component of the acceleration at $$t = 2.00 s$$?

Accepted Answer

$$a_x = 10\ \mathrm{m/s^2$}$$

Question 14

If two people pull on a $$209\ \mathrm{kg}$$ boat in opposite directions and the acceleration is $$0.518\ \mathrm{m/s^2$} to $$the right, what is the net force acting on the boat?

Accepted Answer

$$F_{net} = 209 	imes 0.518 = 108.2\ \mathrm{N}$$

Question 15

A force $$F is $$applied to two masses $$m_1$$ and $$m_2$$ connected by a string on a frictionless surface. If $$m_1 = 2$$\ \mathrm{kg}$$, m_2$ = 3\ \mathrm{kg}$$, and $$F = 10\ \mathrm{N}$$, what is the tension in the string?

Accepted Answer

$$T = \frac{m_1$ F}{m_1$ + m_2$} = \frac{2 	imes 10}{5} = 4\ \mathrm{N}$$

Newton's Laws, Forces, and Equilibrium: Homework Problems and Applications

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