Two forces act on a 55-kg object. One force has the magnitude 65 N directed 59 clockwise from the positive x-axis, and the other has a magnitude ofm35 N at 32 clockwise from the positive y axis. What is the magnitude of acceleration?
1.1
To find the magnitude of acceleration, we need to calculate the net force acting on the object first.
Let's break down the given forces into their x and y components:
Force 1:
Magnitude = 65 N
Angle = 59 degrees clockwise from the positive x-axis
x-component = 65 N * cos(59°)
y-component = 65 N * sin(59°)
Force 2:
Magnitude = 35 N
Angle = 32 degrees clockwise from the positive y-axis
x-component = 35 N * sin(32°)
y-component = 35 N * cos(32°)
To find the net force, we need to calculate the sum of the x-components and y-components separately.
Net x-component = x-component of Force 1 + x-component of Force 2
Net y-component = y-component of Force 1 + y-component of Force 2
Finally, the magnitude of the net force is given by the formula:
Magnitude of net force = sqrt((Net x-component)^2 + (Net y-component)^2)
Let's plug in the values and calculate:
x-component of Force 1 = 65 N * cos(59°) ≈ 30.03 N
y-component of Force 1 = 65 N * sin(59°) ≈ 55 N
x-component of Force 2 = 35 N * sin(32°) ≈ 18.72 N
y-component of Force 2 = 35 N * cos(32°) ≈ 29.79 N
Net x-component = 30.03 N + 18.72 N ≈ 48.75 N
Net y-component = 55 N + 29.79 N ≈ 84.79 N
Magnitude of net force = sqrt((48.75 N)^2 + (84.79 N)^2) ≈ 97.04 N
Now, to calculate the magnitude of acceleration, we can use Newton's second law: F = ma, where F is the net force and m is the mass of the object.
So, rearranging the equation: a = F / m
Given mass, m = 55 kg
Net force, F = 97.04 N
Magnitude of acceleration = F / m = 97.04 N / 55 kg ≈ 1.76 m/s²
Therefore, the magnitude of acceleration is approximately 1.76 m/s².
To determine the magnitude of acceleration, you need to find the resultant force acting on the object, and then use Newton's second law of motion: F = ma, where F is the resultant force, m is the mass of the object, and a is the acceleration.
To find the resultant force, you can break down the two forces into their x and y components using trigonometry.
Given:
- Force 1: Magnitude = 65 N, Angle = 59° clockwise from the positive x-axis.
- Force 2: Magnitude = 35 N, Angle = 32° clockwise from the positive y-axis.
- Mass of the object: m = 55 kg.
Step 1: Resolve the forces into x and y components.
Force 1:
- F1x = 65 N * cos(59°)
- F1y = 65 N * sin(59°)
Force 2:
- F2x = 35 N * sin(32°)
- F2y = 35 N * cos(32°)
Step 2: Calculate the resultant force components.
- Rx = F1x + F2x
- Ry = F1y + F2y
Step 3: Calculate the magnitude of the resultant force.
- R = sqrt(Rx^2 + Ry^2)
Step 4: Calculate the acceleration using Newton's second law.
- a = R / m
Now, let's calculate the values:
Step 1:
F1x = 65 N * cos(59°) = 33.54 N
F1y = 65 N * sin(59°) = 54.67 N
F2x = 35 N * sin(32°) = 18.04 N
F2y = 35 N * cos(32°) = 29.53 N
Step 2:
Rx = F1x + F2x = 33.54 N + 18.04 N = 51.58 N
Ry = F1y + F2y = 54.67 N + 29.53 N = 84.2 N
Step 3:
R = sqrt(Rx^2 + Ry^2) = sqrt((51.58 N)^2 + (84.2 N)^2) = sqrt(2661.96 + 7089.64) = 93.15 N
Step 4:
a = R / m = 93.15 N / 55 kg = 1.69 m/s^2
Therefore, the magnitude of acceleration is 1.69 m/s^2.