How far x does it move down the incline before coming to rest? Suppose the incline is frictionless. The angle of inclination is 36 ◦ , the spring constant is 89.6 N/m the mass of the block is 9.05 kg and the acceleration of gravity is 9.8 m/s2 .

Answer in units of m
Choosing down the incline as the positive direction, what is its acceleration at its lowest point?
Answer in units of m/s2

weight component down incline = m g sin 36

= 9.05 * 9.8* sin 36
= 52.1 Newtons compressing spring
work done by gravity = energy in spring
52.1 x = (1/2) 89.6 x^2
x = 1.16 m
F spring = 89.6 * 1.16 =104 Newtons up incline
net force up = 104 - 52.1 = 52.1 up
a = F/m = 52.1/9.05 = 5.76 m/s^2

Well, well, well, we have an incline, a spring, and some mathematical conundrums. It seems like we are in for a fun ride!

Now, let's get down to business. We have a frictionless incline with an angle of 36 degrees, a spring constant of 89.6 N/m, a block with a mass of 9.05 kg, and gravity giving us a helping hand at 9.8 m/s².

To determine how far our brave little block moves down the incline before coming to rest, we need to first calculate the gravitational force pulling it in that direction. This force can be found by multiplying the mass of the block by the sine of the angle of inclination (36 degrees).

F_gravity = mass * gravity * sin(angle)

F_gravity = 9.05 kg * 9.8 m/s² * sin(36)

Now that we know the gravitational force, we can determine the maximum force that the spring can provide. This force is equal to the extension of the spring times the spring constant (Hooke's Law).

F_spring = k * x

Here, x represents how far the block moves down the incline before coming to rest. We're looking for that little fella! Now solving for x, we have:

F_gravity = F_spring

mass * gravity * sin(angle) = k * x

9.05 kg * 9.8 m/s² * sin(36) = 89.6 N/m * x

Solving for x, we find:

x = (9.05 kg * 9.8 m/s² * sin(36)) / 89.6 N/m

So, the block moves approximately x = 0.8572 meters down the incline before coming to rest. Phew!

Now, moving on to the next question. Since we are choosing down the incline as the positive direction, and the block is coming to rest, we can conclude that the acceleration at its lowest point is zero. Zip, nada, zilch! No acceleration to be found there!

Hope I managed to bring a smile to your face while solving these physics problems. Remember, laughter is the best reaction!

To determine how far the block moves down the incline before coming to rest, we can use conservation of mechanical energy. At the highest point, the block only has potential energy. At the lowest point, the potential energy is converted entirely to kinetic energy and spring potential energy.

The potential energy at the highest point is given by:

PE = m * g * h

where m is the mass of the block, g is the acceleration due to gravity, and h is the height of the incline.

The height of the incline can be determined from the given angle of inclination, θ. Using trigonometry, we can calculate:

h = x * sin(θ)

where x is the distance the block moves down the incline.

The potential energy at the highest point is then:

PE = m * g * x * sin(θ)

At the lowest point, all the potential energy is converted to kinetic energy and spring potential energy. The kinetic energy is given by:

KE = (1/2) * m * v^2

where v is the velocity at the lowest point.

The spring potential energy is given by:

SPE = (1/2) * k * x^2

where k is the spring constant.

Since the block comes to rest at the lowest point, its final velocity, v, is 0. This means that the kinetic energy is also 0.

Setting up the energy conservation equation:

PE = KE + SPE

m * g * x * sin(θ) = 0 + (1/2) * k * x^2

Simplifying and rearranging the equation:

m * g * x * sin(θ) = (1/2) * k * x^2

x * sin(θ) = (1/2) * (k / (m * g)) * x^2

Simplifying further:

sin(θ) = (k / (2 * m * g)) * x

x = (2 * m * g * sin(θ)) / k

Plugging in the given values:

x = (2 * 9.05 kg * 9.8 m/s^2 * sin(36°)) / 89.6 N/m

Calculating this expression, we find:

x ≈ 0.995 m

Therefore, the block moves approximately 0.995 meters down the incline before coming to rest.

To determine the acceleration at the lowest point, we can use Newton's second law:

F = m * a

where F is the net force acting on the block and a is its acceleration.

At the lowest point, the net force acting on the block is the force due to gravity component along the incline. It can be calculated as:

F = m * g * sin(θ)

Substituting the given values:

F = 9.05 kg * 9.8 m/s^2 * sin(36°)

Calculating this expression, we find:

F ≈ 50.66 N

Since the mass of the block does not change, the acceleration is given by:

a = F / m

Substituting the values:

a = 50.66 N / 9.05 kg

Calculating this expression, we find:

a ≈ 5.603 m/s^2

Therefore, the acceleration at the lowest point is approximately 5.603 m/s^2.