An crate weighing 510 N is resting on a plane inclined 30° above the horizontal.

(a) Calculate the magnitude of the acceleration (ignore friction).


m/s2

(b) After 3.00 s, how fast will the crate be moving?

component of gravity down slope = 9.81 sin 30 = .5(9.81)

F = m a

m (.5)(9.81) = m a

a = 9.81/2 = 4.9 m/s^2 approx

after 3
v = a t = 4.9*3 = 14.7 m/s

(a) Well, first things first, let's find an excuse for the crate to get moving. Maybe it's being chased by a herd of angry squirrels or it heard a rumor that gravity is more fun on the other side of the inclined plane. Anyway, to find the acceleration, we can use the good old formula:

acceleration = (force net) / (mass)

In this case, the only force acting on the crate is its weight. So, let's calculate it:

force net = mass * acceleration due to gravity

The mass of the crate doesn't seem to be given, but since we have its weight (510 N), we can divide that by the acceleration due to gravity (9.8 m/s^2) to get the mass:

mass = 510 N / 9.8 m/s^2 = 52.04 kg

Now we can finally calculate the acceleration:

acceleration = (force net) / (mass) = 510 N / 52.04 kg = 9.8 m/s^2

(b) After 3.00 s, the crate will be moving with a velocity that is equal to the acceleration multiplied by the time:

velocity = acceleration * time = 9.8 m/s^2 * 3.00 s = 29.4 m/s

So, the crate will be moving at a speed of 29.4 m/s. Just make sure it doesn't run into any squirrels along the way!

To solve this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.

(a) Let's calculate the magnitude of the acceleration:

First, we need to resolve the force of gravity into its components parallel and perpendicular to the inclined plane. The force of gravity acting on the crate can be split into two components:

F_parallel = m * g * sin(θ)
F_perpendicular = m * g * cos(θ)

where:
m = mass of the crate
g = acceleration due to gravity = 9.8 m/s^2
θ = angle of inclination = 30°

Since the crate is at rest, the only force acting parallel to the inclined plane is F_parallel. Therefore, F_parallel will be the net force acting on the crate.

F_parallel = m * a

Substituting the known values, we have:

m * g * sin(θ) = m * a

Canceling out the mass (m) on both sides of the equation, we get:

g * sin(θ) = a

Substituting the values, we have:

a = 9.8 m/s^2 * sin(30°)
a ≈ 4.9 m/s^2

Therefore, the magnitude of the acceleration is approximately 4.9 m/s^2.

(b) To find the velocity after 3.00 s, we can use the formula:

v = u + at

where:
v = final velocity
u = initial velocity (in this case, 0 m/s because the crate is at rest)
a = acceleration (calculated in part a)
t = time = 3.00 s

Substituting the known values, we have:

v = 0 + (4.9 m/s^2) * (3.00 s)
v = 14.7 m/s

Therefore, after 3.00 s, the crate will be moving at a speed of 14.7 m/s.

To calculate the magnitude of the acceleration, we can use the components of the weight force acting on the crate.

Step 1: Find the weight force acting on the crate.
The weight force is given by W = mg, where m is the mass of the crate and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Given that the weight force is 510 N, we can rearrange the formula to solve for the mass:
m = W/g = 510 N / 9.8 m/s^2 = 52.04 kg.

Step 2: Resolve the weight force into its components.
The weight force can be split into two components: one parallel to the inclined plane (mg*sinθ) and another perpendicular to the inclined plane (mg*cosθ), where θ is the angle of inclination.

For the parallel component (F_parallel), we have F_parallel = mg*sinθ.
F_parallel = (52.04 kg) * (9.8 m/s^2) * sin(30°) = 254.89 N.

For the perpendicular component (F_perpendicular), we have F_perpendicular = mg*cosθ.
F_perpendicular = (52.04 kg) * (9.8 m/s^2) * cos(30°) = 445.39 N.

Step 3: Calculate the acceleration.
Since there is no friction mentioned in the question, the net force acting on the crate is equal to the parallel component of the weight force:
net force = F_parallel = 254.89 N.

Using Newton's second law (F = ma), we can find the acceleration:
a = net force / mass = 254.89 N / 52.04 kg ≈ 4.90 m/s².

Therefore, the magnitude of the acceleration is approximately 4.90 m/s².

(b) To find the velocity of the crate after 3.00 seconds, we can use the kinematic equation:
v = u + at,

where
v = final velocity,
u = initial velocity (assumed to be 0 as the crate is initially at rest),
a = acceleration, and
t = time.

Given that the acceleration is 4.90 m/s² and the time is 3.00 s, we can substitute these values:
v = 0 + (4.90 m/s²) * (3.00 s) = 14.70 m/s.

Therefore, after 3.00 seconds, the crate will be moving at a speed of 14.70 m/s.