Q: The sun ($$2 \times 10$$^{30}$$\ \mathrm{kg}) $$and Jupiter ($$1.9 \times 10$$^{27}$$\ \mathrm{kg}) $$are $$7.8 \times 10^8$\ \mathrm{km}$$ apart. What is the approximate distance from the center of the sun to the center of mass of the system?

$$7.4 \times 10^5$\ \mathrm{km}$$

Question 1

Which of the following best describes the center of mass of a system of particles?

Accepted Answer

It is the weighted average position of all particles, weighted by their masses.

Question 2

Given three masses: $$m_A = 300\ \mathrm{g} at$$ $$(0,0)$$, $$m_B = 200\ \mathrm{g} at$$ $$(0,10\ \mathrm{cm})$$, and $$m_C = 100\ \mathrm{g} at$$ $$(10\ \mathrm{cm}, 0)$$, what are the coordinates of the center of mass?

Accepted Answer

$$(5\ \mathrm{cm}, 3.3\ \mathrm{cm})$$

Question 3

A force $$F is $$applied at a distance $$L$$ from a pivot at an angle $$	heta to $$the lever arm. What is the magnitude of the torque produced?

Accepted Answer

$$	au = FL\sin	heta$$

Question 4

The sun ($$2 	imes 10$$^{30}$$\ \mathrm{kg}) $$and Jupiter ($$1.9 	imes 10$$^{27}$$\ \mathrm{kg}) $$are $$7.8 	imes 10^8$\ \mathrm{km}$$ apart. What is the approximate distance from the center of the sun to the center of mass of the system?

Accepted Answer

$$7.4 	imes 10^5$\ \mathrm{km}$$

Question 5

Which statement about the rigid-body model is correct?

Accepted Answer

It assumes the object's size and shape do not change as it moves.

Question 6

A uniform meter stick of mass $$150\ \mathrm{g}$$ has masses $$m_1 = 50$$\ \mathrm{g}$$ at 30$$\ \mathrm{cm}$$ and m_2$ = 75\ \mathrm{g} at$$ $$70\ \mathrm{cm}. $$Where should a $$100\ \mathrm{g}$$ mass be placed to balance the stick at its center?

Accepted Answer

$$80\ \mathrm{cm}$$

Question 7

A person pushes a door at its edge, perpendicular to the surface. If the force is doubled and the distance from the hinge is halved, what happens to the torque?

Accepted Answer

It remains the same.

Question 8

A 2 kg rod, 2 m long, leans against a frictionless wall at $$60^\circ$$ with the ground. What is the minimum coefficient of static friction $$\mu_s$$ required to prevent slipping if a 100 kg person stands at the top?

Accepted Answer

$$\mu_s = 0.87$$

Question 9

Which of the following is true about static equilibrium?

Accepted Answer

The net force and net torque on the object must both be zero.

Question 10

A sign of mass $$2\ \mathrm{kg}$$ hangs from a $$2\ \mathrm{m}$$ long, $$5\ \mathrm{kg}$$ rod mounted at one end. The sign is attached $$0.5\ \mathrm{m}$$ from the right end. What is the tension in the supporting rope if the system is in equilibrium?

Accepted Answer

$$T = 29.4\ \mathrm{N}$$

Question 11

If a force is applied at an angle $$	heta to a $$lever arm, which value of $$	heta$$ produces the maximum torque?

Accepted Answer

$$	heta = 90^\circ$$

Question 12

A wrench is used to loosen a bolt. Which action increases the torque most effectively?

Accepted Answer

Applying a greater force farther from the bolt, perpendicular to the wrench.

Question 13

Which of the following is NOT a limitation of the rigid-body model?

Accepted Answer

It cannot describe changes in shape.

Question 14

A uniform beam is supported horizontally by two supports at its ends. If a heavy object is placed closer to one end, what happens to the forces at the supports?

Accepted Answer

The support closer to the object bears more force.

Question 15

Which equation correctly gives the $$x-$$coordinate of the center of mass for $$n$$ particles?

Accepted Answer

$$x_{cm} = \frac{m_1x_1 + m_2x_2 + \ldots + m_nx_n}{m_1$ + m_2$ + \ldots + m_n}$$

Question 16

A ladder leans against a wall at $$60^\circ$$ with the ground. If the ladder is $$2\ \mathrm{m}$$ long and $$2\ \mathrm{kg}$$, what is the normal force from the ground when a $$100\ \mathrm{kg}$$ person stands at the top?

Accepted Answer

$$N = 1002\ \mathrm{N}$$

Question 17

Which of the following scenarios is NOT in static equilibrium?

Accepted Answer

A spinning wheel with constant angular velocity.

Question 18

A beam is balanced on a fulcrum. If the mass on one side is doubled and the distance from the fulcrum is halved, what happens to the torque on that side?

Accepted Answer

It remains the same.

Question 19

Which of the following best explains why particles of higher mass count more in the center of mass calculation?

Accepted Answer

Because heavier particles contribute more to the total mass and thus shift the center of mass toward themselves.

Extended Bodies at Rest: Center of Mass, Torque, and Static Equilibrium

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