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
damon your answer is wrong i hate you
Someone please answer i need help and do not understand
Damon, I love you. By some miracle, my problem had the exact same numbers as Alex so my answers were the same as yours (and I didn't have to do work). The answer for acceleration is negative, not positive, for some reason. If only Grant could've tried changing the sign, maybe next time bud
Damon, I love you. I had different numbers but I just substituted Alex's numbers with mine and it worked.
To determine how far the object moves down the incline before coming to rest, we can use the principles of conservation of mechanical energy. At its highest point, the object has only gravitational potential energy, and at its lowest point, it has both gravitational potential energy and elastic potential energy from the compressed spring.
To find the distance x, we need to equate the initial gravitational potential energy at the highest point to the sum of gravitational potential and elastic potential energy at the lowest point.
Given:
- Angle of inclination (θ) = 36 degrees
- Spring constant (k) = 89.6 N/m
- Mass of the block (m) = 9.05 kg
- Acceleration due to gravity (g) = 9.8 m/s^2
Let's calculate the distance x first.
To calculate x, we need to use the equation for gravitational potential energy:
Gravitational Potential Energy = m * g * h
where
m = mass of the object
g = acceleration due to gravity
h = height of the incline
Since the incline is frictionless, there is no energy loss due to friction, and the gravitational potential energy at the highest point is equal to the sum of gravitational potential energy and the elastic potential energy at the lowest point.
At the highest point, the entire potential energy is in the form of gravitational potential energy (as there is no spring compression), so:
m * g * h = (1/2) * k * x^2
Solving for x:
x^2 = (2 * m * g * h) / k
x = sqrt((2 * m * g * h) / k)
Substituting the given values,
x = sqrt((2 * 9.05 kg * 9.8 m/s^2 * h) / (89.6 N/m))
Now, let's calculate the acceleration at the lowest point.
To find the acceleration, we can use Newton's second law of motion, which states that the sum of the forces acting on an object is equal to the mass times the acceleration.
Here, the forces acting on the object are the gravitational force and the force due to the spring.
Gravitational force = m * g * sin(θ)
Spring force = k * x
Since the incline is frictionless, there are no other forces at play.
The net force acting on the object is given by:
Net force = Gravitational force - Spring force
ma = m * g * sin(θ) - k * x
Simplifying,
a = (g * sin(θ) - (k / m) * x)
Now let's calculate the acceleration using the values provided.
a = (9.8 m/s^2 * sin(36 degrees)) - (89.6 N/m / 9.05 kg * x))
Remember to convert the angle to radians before taking its sine.
Finally, substitute the calculated value of x into the equation to find the acceleration.
Please provide the value of 'h' (the height of the incline) to complete the calculation.