Q: A force $$\vec{F}$$ acts on a particle moving in cylindrical coordinates. Which is the correct expression for infinitesimal work?

$$\delta U = F_r dr + F_\theta r d\theta + F_z dz$$

Question 1

Which of the following correctly expresses the infinitesimal work $$\delta U$$ done by a force $$\vec{F}$$ during a displacement $$d\vec{r}$$?

Accepted Answer

$$\delta U = \vec{F} \cdot d\vec{r}$$

Question 2

A block of mass $$m is $$lifted from height $$y_1 to$$ $$y_2 by a $$force equal to its weight. What is the work done by gravity during this process?

Accepted Answer

$$-mg(y_2$ - y_1$)$$

Question 3

Which statement about conservative forces is correct?

Accepted Answer

The work done by a conservative force depends only on the initial and final positions, not the path taken.

Question 4

For a spring with force $$\vec{F} = -kx\hat{x}$$, what is the total work done by the spring as it moves from $$x_1 to$$ $$x_2$$?

Accepted Answer

$$\frac{k}{2}(x_1$^2 - x_2$^2)$$

Question 5

A force $$\vec{F}$$ acts on a particle moving with velocity $$\vec{v}. $$What is the instantaneous power delivered by the force?

Accepted Answer

$$P = \vec{F} \cdot \vec{v}$$

Question 6

A 0.5-kg collar slides along a spiral rod $$r = 0.3	heta$$ under a constant radial force $$T = 10 N. If $$released from rest at $$	heta = \frac{\pi}{2}$$ and slides to $$	heta = \pi$$, what is the velocity $$v at$$ $$B$$?

Accepted Answer

$$v = \sqrt{\frac{\alpha \pi T}{m}}$$

Question 7

Which of the following is the correct expression for kinetic energy $$K in $$Cartesian coordinates?

Accepted Answer

$$K = \frac{m}{2}(\dot{x}^2 + \dot{y}^2 + \dot{z}^2)$$

Question 8

The work-energy theorem states that:

Accepted Answer

The work done by the resultant force equals the change in kinetic energy.

Question 9

A block slides down a frictionless incline. Which statement is true about the work done by gravity?

Accepted Answer

It is equal to the change in the block's kinetic energy.

Question 10

If a force is perpendicular to the displacement, what is the work done?

Accepted Answer

Zero

Question 11

A spring with constant $$k is $$compressed from $$x = 0 to$$ $$x = a. $$What is the work done by the spring?

Accepted Answer

$$-\frac{k}{2}a^2$

Question 12

Which of the following best describes power in physics?

Accepted Answer

The rate at which work is done

Question 13

A force $$\vec{F}$$ acts on a particle moving in cylindrical coordinates. Which is the correct expression for infinitesimal work?

Accepted Answer

$$\delta U = F_r dr + F_	heta r d	heta + F_z dz$$

Question 14

A car engine delivers a constant power $$P to $$accelerate a car of mass $$m$$ from rest. Ignoring friction, what is the car's speed $$v$$ after time $$t$$?

Accepted Answer

$$v = \sqrt{\frac{2Pt}{m}}$$

Question 15

A block is pushed horizontally with a constant force $$F$$ over a distance $$L at an $$angle $$\alpha to $$the direction of motion. What is the work done?

Accepted Answer

$$F L \cos(\alpha)$$

Question 16

Which of the following is NOT a condition for zero work?

Accepted Answer

Force is parallel to displacement

Question 17

A resistive force does negative work. Which scenario best illustrates this?

Accepted Answer

A car braking to a stop

Question 18

If a particle moves under a force $$\vec{F}$$ and its velocity is $$\vec{v}$$, which equation relates the change in kinetic energy to the work done?

Accepted Answer

$$K_2$$ - $$K_1 = U$$_{1 \rightarrow 2}(\vec{R})$$

Question 19

A block is moved horizontally by a force $$F$$ for a distance $$L. If $$the force is perpendicular to the displacement, what is the work done?

Accepted Answer

$$0$$

Work, Power, and the Work-Energy Theorem in Physics

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