To calculate the work done by each of the three forces ($T_1$, $T_2$, and $T_3$) on the crate as it moves 4.6 m across the floor, we use the formula for work:

$W = F \cdot d \cdot \cos(\theta)$

where:

• $W$ is the work done by the force,
• $F$ is the magnitude of the force,
• $d$ is the distance over which the force is applied (in this case, 4.6 m),
• $\theta$ is the angle between the force and the direction of motion.

Analyzing Each Force:

1. Force $T_1 = 600 \, \text{N}$:

   • The angle $\theta_1$ between $T_1$ and the direction of motion is $20^\circ$.
   • Work done by $T_1$: $W_1 = 600 \cdot 4.6 \cdot \cos(20^\circ)$

2. Force $T_2 = 410 \, \text{N}$:

   • The angle $\theta_2$ between $T_2$ and the direction of motion is $30^\circ$.
   • Work done by $T_2$: $W_2 = 410 \cdot 4.6 \cdot \cos(30^\circ)$

3. Force $T_3 = 660 \, \text{N}$:

   • The angle $\theta_3$ between $T_3$ and the direction of motion is $180^\circ$, indicating that $T_3$ is directly opposite to the motion.
   • Work done by $T_3$: $W_3 = 660 \cdot 4.6 \cdot \cos(180^\circ)$
   • Since $\cos(180^\circ) = -1$, $W_3$ will be negative, indicating that $T_3$ does work against the motion.

Calculating the Values:

Now, let’s compute each component.The work done by each of the three forces is as follows:

1. Work done by $T_1$: $W_1 = 2593.55 \, \text{J}$
2. Work done by $T_2$: $W_2 = 1633.32 \, \text{J}$
3. Work done by $T_3$: $W_3 = -3036.00 \, \text{J}$ (negative, indicating work against the direction of motion)

Would you like further explanation on any of these calculations, or do you have additional questions?

Here are 5 related questions to consider:

1. How does the direction of a force affect the work done?
2. What would happen to the work values if the crate moved a longer distance?
3. How would changing the angles of the forces impact the work done?
4. Why is the work by $T_3$ negative, and what does it imply?
5. How could we determine the net work done on the crate?

Tip: Remember, work is positive if the force is in the direction of motion and negative if it opposes it.