Three forces acting on an object are given by F1 = ( 1.95i - 1.60j ) N, F2 = ( - 4.50i - 3.25j ) N, and F3 = ( 46.0j ) N. The object experiences an acceleration of magnitude 3.85 m/s2
(a) What is the direction of the acceleration?
(b) What is the mass of the object?
(c) What are the velocity components of the object after 19.0 s?
(a) The direction of the acceleration is the direction of the sum of those three vectors. Add them up. The ratio of the i and j sums will tell you the direction of the acceleration. You will need the components and the magnitude for later questions.
(b) The mass is the net force magnitude divided by the acceleration.
(c) The velocity components will be the acceleration components (net force vector divided by mass) multiplied by 19 seconds.
To find the answers to these questions, we can use Newton's second law of motion, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. We can break down the problem into several steps:
Step 1: Find the net force acting on the object.
Step 2: Determine the direction of the acceleration based on the net force.
Step 3: Calculate the mass of the object using the formula.
Step 4: Use the acceleration and time to find the final velocity components.
Let's begin with step 1:
Step 1: Find the net force acting on the object.
The net force is the vector sum of all the forces acting on the object. We can calculate this by summing the individual force vectors:
Net force (Fnet) = F1 + F2 + F3
F1 = (1.95i - 1.60j) N
F2 = (-4.50i - 3.25j) N
F3 = (0i + 46.0j) N
Adding these vectors together, we get:
Fnet = F1 + F2 + F3
Fnet = (1.95i - 1.60j) N + (-4.50i - 3.25j) N + (0i + 46.0j) N
Fnet = (-2.55i + 41.15j) N
Step 2: Determine the direction of the acceleration based on the net force.
The direction of the acceleration is the same as the direction of the net force. In this case, the net force is given by (-2.55i + 41.15j) N. Therefore, the direction of the acceleration is in the positive y-direction.
Step 3: Calculate the mass of the object using the formula.
Newton's second law of motion states that Fnet = m * a, where Fnet is the net force, m is the mass of the object, and a is the acceleration.
Using the given magnitude of the acceleration (3.85 m/s^2) and the net force we calculated earlier, we can solve for mass:
Fnet = m * a
(-2.55i + 41.15j) N = m * (3.85 m/s^2)
Since the force and acceleration are in the y-direction, we can focus on the y-component of the equation:
41.15j N = m * (3.85 m/s^2)
Comparing the y-components, we find:
41.15 N = m * (3.85 m/s^2)
Solving for mass (m), we divide both sides by 3.85 m/s^2:
m = 41.15 N / 3.85 m/s^2
m ≈ 10.68 kg
Step 4: Use the acceleration and time to find the final velocity components.
To find the velocity components after 19.0 s, we can use the formula v = u + at, where v is the final velocity, u is the initial velocity (which can be assumed to be zero in this case), a is the acceleration, and t is the time.
In this case, the acceleration is given as 3.85 m/s^2 and the time is 19.0 s.
Using the formula, we can calculate the velocity components:
vx = 0 + (1.95 m/s^2) * 19.0 s = 37.05 m/s (in the x-direction)
vy = 0 + (41.15 m/s^2) * 19.0 s = 781.85 m/s (in the y-direction)
Therefore, the velocity components of the object after 19.0 s are approximately 37.05 m/s in the x-direction and 781.85 m/s in the y-direction.
To summarize:
(a) The direction of the acceleration is in the positive y-direction.
(b) The mass of the object is approximately 10.68 kg.
(c) The velocity components of the object after 19.0 s are approximately 37.05 m/s in the x-direction and 781.85 m/s in the y-direction.