Answer :
To solve the problem, let's break it down step-by-step.
We need to find the components of the acceleration [tex]\( a_x \)[/tex] and [tex]\( a_y \)[/tex] for a mass of 3 kg when a force of 12 N is applied at an angle of 55 degrees relative to the positive x-axis.
### Step 1: Understand the given data
- Mass (m) = 3 kg
- Force (F) = 12 N
- Angle ([tex]\(\theta\)[/tex]) = 55 degrees
### Step 2: Break the force into its x and y components
To find the components of the force in the x and y directions, we use the following formulas:
- [tex]\( F_x = F \cdot \cos(\theta) \)[/tex]
- [tex]\( F_y = F \cdot \sin(\theta) \)[/tex]
### Step 3: Find acceleration components using Newton's Second Law
Newton's Second Law is [tex]\( F = m \cdot a \)[/tex], where [tex]\( a \)[/tex] is the acceleration. We can rearrange this to find acceleration as:
- [tex]\( a_x = \frac{F_x}{m} \)[/tex]
- [tex]\( a_y = \frac{F_y}{m} \)[/tex]
### Step 4: Calculate the x-component of the acceleration
First, we find the cosine of the angle to determine [tex]\( a_x \)[/tex]:
- [tex]\( \cos(55^\circ) = 0.573576436351046 \)[/tex]
Next, calculate the force in the x direction:
- [tex]\( F_x = 12 \, \text{N} \times 0.573576436351046\)[/tex]
- [tex]\( F_x \approx 6.882917236212552 \, \text{N} \)[/tex]
Now, find the acceleration in the x direction:
- [tex]\( a_x = \frac{6.882917236212552 \, \text{N}}{3 \, \text{kg}} \)[/tex]
- [tex]\( a_x \approx 2.294 \, \text{m/s}^2 \)[/tex]
Thus, the x-component of the object's acceleration is approximately [tex]\( 2.294 \, \text{m/s}^2 \)[/tex].
### Step 5: Calculate the y-component of the acceleration
Next, we find the sine of the angle to determine [tex]\( a_y \)[/tex]:
- [tex]\( \sin(55^\circ) = 0.819152044288992 \)[/tex]
Next, calculate the force in the y direction:
- [tex]\( F_y = 12 \, \text{N} \times 0.819152044288992 \)[/tex]
- [tex]\( F_y \approx 9.829824531467904 \, \text{N} \)[/tex]
Now, find the acceleration in the y direction:
- [tex]\( a_y = \frac{9.829824531467904 \, \text{N}}{3 \, \text{kg}} \)[/tex]
- [tex]\( a_y \approx 3.277 \, \text{m/s}^2 \)[/tex]
Therefore, the y-component of the object's acceleration is approximately [tex]\( 3.277 \, \text{m/s}^2 \)[/tex].
### Summary of Components:
- The x-component of the acceleration is [tex]\( 2.294 \, \text{m/s}^2 \)[/tex] (Choice A).
- The y-component of the acceleration is [tex]\( 3.277 \, \text{m/s}^2 \)[/tex] (Choice D).
So, the correct answers are:
- x-component: [tex]\( \boxed{2.294 \, \text{m/s}^2} \)[/tex]
- y-component: [tex]\( \boxed{3.277 \, \text{m/s}^2} \)[/tex]
We need to find the components of the acceleration [tex]\( a_x \)[/tex] and [tex]\( a_y \)[/tex] for a mass of 3 kg when a force of 12 N is applied at an angle of 55 degrees relative to the positive x-axis.
### Step 1: Understand the given data
- Mass (m) = 3 kg
- Force (F) = 12 N
- Angle ([tex]\(\theta\)[/tex]) = 55 degrees
### Step 2: Break the force into its x and y components
To find the components of the force in the x and y directions, we use the following formulas:
- [tex]\( F_x = F \cdot \cos(\theta) \)[/tex]
- [tex]\( F_y = F \cdot \sin(\theta) \)[/tex]
### Step 3: Find acceleration components using Newton's Second Law
Newton's Second Law is [tex]\( F = m \cdot a \)[/tex], where [tex]\( a \)[/tex] is the acceleration. We can rearrange this to find acceleration as:
- [tex]\( a_x = \frac{F_x}{m} \)[/tex]
- [tex]\( a_y = \frac{F_y}{m} \)[/tex]
### Step 4: Calculate the x-component of the acceleration
First, we find the cosine of the angle to determine [tex]\( a_x \)[/tex]:
- [tex]\( \cos(55^\circ) = 0.573576436351046 \)[/tex]
Next, calculate the force in the x direction:
- [tex]\( F_x = 12 \, \text{N} \times 0.573576436351046\)[/tex]
- [tex]\( F_x \approx 6.882917236212552 \, \text{N} \)[/tex]
Now, find the acceleration in the x direction:
- [tex]\( a_x = \frac{6.882917236212552 \, \text{N}}{3 \, \text{kg}} \)[/tex]
- [tex]\( a_x \approx 2.294 \, \text{m/s}^2 \)[/tex]
Thus, the x-component of the object's acceleration is approximately [tex]\( 2.294 \, \text{m/s}^2 \)[/tex].
### Step 5: Calculate the y-component of the acceleration
Next, we find the sine of the angle to determine [tex]\( a_y \)[/tex]:
- [tex]\( \sin(55^\circ) = 0.819152044288992 \)[/tex]
Next, calculate the force in the y direction:
- [tex]\( F_y = 12 \, \text{N} \times 0.819152044288992 \)[/tex]
- [tex]\( F_y \approx 9.829824531467904 \, \text{N} \)[/tex]
Now, find the acceleration in the y direction:
- [tex]\( a_y = \frac{9.829824531467904 \, \text{N}}{3 \, \text{kg}} \)[/tex]
- [tex]\( a_y \approx 3.277 \, \text{m/s}^2 \)[/tex]
Therefore, the y-component of the object's acceleration is approximately [tex]\( 3.277 \, \text{m/s}^2 \)[/tex].
### Summary of Components:
- The x-component of the acceleration is [tex]\( 2.294 \, \text{m/s}^2 \)[/tex] (Choice A).
- The y-component of the acceleration is [tex]\( 3.277 \, \text{m/s}^2 \)[/tex] (Choice D).
So, the correct answers are:
- x-component: [tex]\( \boxed{2.294 \, \text{m/s}^2} \)[/tex]
- y-component: [tex]\( \boxed{3.277 \, \text{m/s}^2} \)[/tex]
