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
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
To find out how much the spring stretches with the brick attached, we need to consider the forces acting on the brick on the inclined plane.
First, let's analyze the forces in the vertical direction. Since the ramp is perfectly smooth, there is no friction acting on the brick. Therefore, the only force acting in the vertical direction is the gravitational force, which can be calculated as:
F_gravity = m * g
= 3.0 kg * 9.8 m/s^2
= 29.4 N
Now let's consider the forces in the parallel direction to the ramp. This will help us determine whether the spring stretches or compresses. The force due to gravity can be split into two components: one acting parallel to the ramp (F_parallel) and one perpendicular to the ramp (F_perpendicular).
F_parallel = m * g * sin(θ)
= 3.0 kg * 9.8 m/s^2 * sin(34°)
= 49.7 N
The spring force (F_spring) will be equal in magnitude but opposite in direction to the force parallel to the ramp. So:
F_spring = -F_parallel
= -49.7 N
According to Hooke's Law, the force exerted by a spring is directly proportional to the displacement of the spring. The formula for Hooke's Law is:
F_spring = -k * x
Where F_spring is the spring force, k is the spring constant, and x is the displacement of the spring from its equilibrium position.
Substituting the values we know:
-49.7 N = -120 N/m * x
To solve for x:
x = -49.7 N / -120 N/m
x ≈ 0.414 m
Finally, we convert the displacement to centimeters:
x ≈ 0.414 m * 100 cm/m
x ≈ 41.4 cm
Therefore, the spring stretches approximately 41.4 cm with the brick attached.
None of the given answer choices match exactly with the calculated result of 41.4 cm. However, the closest answer choice is 36 cm (E). Please note that this discrepancy could indicate an error in the answer choices provided.