Q: What is the angular momentum of a particle relative to an origin $$O$$?

$$\vec{L} = \vec{r} \times \vec{p}$$

Question 1

Which equation correctly represents the position vector of the center of mass for a system of particles?

Accepted Answer

$$\vec{r}_{cm} = \frac{\sum_i m_i \vec{r}_i}{\sum_i m_i}$$

Question 2

For a homogeneous cube, sphere, or cylinder, where is the center of mass located?

Accepted Answer

At the geometric center of the object

Question 3

If a disk and a donut both have an axis of symmetry, where does the center of mass lie?

Accepted Answer

Along the axis of symmetry, but for the donut it may not be within the object

Question 4

A wheel rolls without slipping. What is the relationship between the velocity of the center of mass $$v_{cm}$$ and the angular speed $$\omega$$?

Accepted Answer

$$v_{cm} = R\omega$$

Question 5

Which equation gives the total kinetic energy of a rigid body undergoing both translation and rotation?

Accepted Answer

$$K = \frac{1}{2} M v_{cm}^2 + \frac{1}{2} I_{cm} \$$omega^2$$

Question 6

A drag racer’s tires are slipping on the road. Which statement is true?

Accepted Answer

The velocity of the center of mass $$v_{cm} is $$not equal to $$R\omega$$

Question 7

Which equation correctly represents the momentum of a particle?

Accepted Answer

$$\vec{p} = m \vec{v}$$

Question 8

What is the angular momentum of a particle relative to an origin $$O$$?

Accepted Answer

$$\vec{L} = \vec{r} 	imes \vec{p}$$

Question 9

Under what condition is the angular momentum of a system conserved?

Accepted Answer

When no net external torque acts on the system

Question 10

Which equation represents the first condition for equilibrium?

Accepted Answer

$$\sum \vec{F} = 0$$

Question 11

Which equation represents the second condition for equilibrium?

Accepted Answer

$$\sum \vec{	au} = 0$$

Question 12

A ladder leans against a frictionless wall. Which force prevents the ladder from slipping?

Accepted Answer

The friction force at the base of the ladder

Question 13

A uniform plank is supported at one end by the floor and at the other by a rope. A person stands three-fourths of the way along the plank. To solve for the tension in the rope, which principle should be used?

Accepted Answer

Sum of torques about the point where the plank touches the floor

Question 14

If the net torque on an object is zero, which statement is true?

Accepted Answer

The object could be both turning and accelerating linearly

Question 15

Which of the following situations satisfies both equilibrium conditions (net force = 0 and net torque = 0)?

Accepted Answer

A seagull gliding at a constant angle below the horizontal and at a constant speed

Question 16

In real-world engineering, why is it important to ensure both equilibrium conditions are met when designing a bridge?

Accepted Answer

To prevent both translational and rotational motion, ensuring structural stability

Question 17

If the center of gravity of an object is not within the object itself, what does this imply about the object's mass distribution?

Accepted Answer

The object has a non-uniform mass distribution or a hollow structure

Question 18

A planet orbits the Sun in a nearly circular path. Which physical quantity is conserved in the absence of external torques?

Accepted Answer

Angular momentum

Question 19

A wheel is rolling without slipping. What is the velocity of the point of contact with the ground?

Accepted Answer

$$v = 0$$

Rolling Motion, Angular Momentum, and Equilibrium – Study Notes

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