A 3.0-kg brick rests on a perfectly smooth ramp inclined at 34° above the horizontal. The brick is kept from sliding down the plane by an ideal spring that is aligned with the surface and attached to a wall above the brick. The spring has a spring constant (force constant) of 120 N/m. By how much does the spring stretch with the brick attached?
A. 24 cm
B.360 cm
C.14 cm
D. 240 cm
E. 36 cm
F = m g sin 34 = 120 x
x = (1/120)(3)(9.8) sin 34
To determine how much the spring stretches with the brick attached, we need to calculate the force exerted by the brick on the spring.
The force exerted by the brick down the inclined plane can be calculated using the equation:
force = mass * acceleration
The acceleration of the brick down the inclined plane can be found using the equation:
acceleration = gravity * sin(angle)
where gravity is the acceleration due to gravity (9.8 m/s²) and angle is the angle of the inclined plane (34°).
Substituting the values into the equation:
acceleration = 9.8 * sin(34°)
Now, let's calculate the force exerted by the brick down the inclined plane:
force = mass * acceleration
force = 3.0 kg * (9.8 * sin(34°))
Next, we need to calculate the force exerted by the spring using Hooke's Law, which states that the force exerted by a spring is proportional to the displacement from its equilibrium position:
force = spring constant * displacement
where the spring constant is given as 120 N/m.
Now we can set up an equation to find the displacement of the spring:
force by the brick = force by the spring
3.0 kg * (9.8 * sin(34°)) = 120 N/m * displacement
Simplifying the equation:
displacement = (3.0 kg * (9.8 * sin(34°))) / 120 N/m
Now, let's calculate the displacement:
displacement = (3.0 * 9.8 * sin(34°)) / 120
Using a calculator, we find that the displacement is approximately 0.24 meters.
Finally, to convert this to centimeters, we multiply by 100:
displacement in centimeters = 0.24 m * 100 = 24 cm
Therefore, the spring stretches by approximately 24 cm with the brick attached.
The answer is A. 24 cm.
To find the amount by which the spring stretches with the brick attached, we need to calculate the gravitational force acting on the brick along the ramp and determine the force exerted by the spring in the opposite direction.
First, let's break down the forces acting on the brick. There are two main forces: gravitational force and the force exerted by the spring.
1. Gravitational force (Fg): The gravitational force acting on the brick can be calculated using the formula Fg = m * g, where m is the mass of the brick and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Fg = 3.0 kg * 9.8 m/s^2 = 29.4 N (approximately)
2. Force exerted by the spring (Fs): The force exerted by the spring can be calculated using Hooke's Law, Fs = k * x, where k is the spring constant (force constant) and x is the displacement (stretch) of the spring.
Here, we need to find x. Given that the ramp is smooth and the brick is not sliding, the force exerted by the spring must be equal in magnitude to the component of the gravitational force parallel to the ramp.
Fs = Fg * sin(θ), where θ is the angle of inclination (34°).
Fs = 29.4 N * sin(34°) = 15.8 N (approximately)
Now, let's find the displacement of the spring, x.
Fs = k * x
15.8 N = 120 N/m * x
x = 15.8 N / 120 N/m = 0.132 m
We have the displacement in meters. To convert it to centimeters, multiply by 100.
Stretch of the spring = 0.132 m * 100 cm/m = 13.2 cm (approximately)
Therefore, the spring stretches by approximately 13.2 cm.
The correct answer is C. 14 cm (which is the closest option).