A block of weight W = 17 N, which can slide without friction on a 32.5° incline, is connected to the top of the incline by a massless spring of unstretched length xo = 0.34 m and spring constant k = 225 N/m. By how much will the spring be stretched when the system is in equilibrium?
If the block is pulled slightly down the incline from its equilibrium position and released, what is the frequency of the ensuing oscillations?
k x = m g sin 32.5
the frequency has nothing to do with gravity or its direction. The stretching and motion and spring force is all parallel to the slope and the gravitational force component is just constant m g sin theta.
omega = sqrt (k/m)
as usual
To find the displacement of the spring when the system is in equilibrium, we need to consider the forces acting on the block on the incline. The weight of the block can be resolved into two components: one parallel to the incline and one perpendicular to it.
1. Determine the component of the weight parallel to the incline:
F_parallel = W * sin(θ)
where θ is the angle of the incline.
Given: W = 17 N and θ = 32.5°
F_parallel = 17 N * sin(32.5°)
2. Calculate the displacement of the spring at equilibrium:
When the system is in equilibrium, the spring force will balance the component of the weight parallel to the incline.
F_spring = -k * (x - xo)
where k is the spring constant, x is the displacement of the spring from its equilibrium position, and xo is the unstretched length of the spring.
Equating the forces:
F_parallel = -F_spring
17 N * sin(32.5°) = -k * (x - 0.34 m)
3. Solve for x, the displacement of the spring:
x = (17 N * sin(32.5°) + (k * 0.34 m)) / k
Now, to find the frequency of the ensuing oscillations when the block is pulled slightly down the incline and released, we can use Hooke's Law:
4. Calculate the effective spring constant:
The effective spring constant when the block is on an inclined plane is given by:
k_eff = k * cos(θ)
where θ is the angle of the incline.
Given: k = 225 N/m and θ = 32.5°
k_eff = 225 N/m * cos(32.5°)
5. Calculate the frequency of oscillations:
The frequency of oscillations is given by:
f = (1/2π) * sqrt(k_eff / m)
where m is the mass of the block.
Given: W = 17 N
m = W / g, where g is the acceleration due to gravity (approximately 9.8 m/s²)
f = (1/2π) * sqrt(k_eff / (W / g))
By following these steps and plugging in the given values, you can find the displacement of the spring at equilibrium and the frequency of oscillations when the block is released.