A puck of mass 4 kg is attached to a spring of force constant 100 N/m and natural length 0.07 meters. The other end of the spring is attached to a fixed pivot. The puck is set into circular motion so that the spring is stretched 0.03 meters from its natural length as the puck circles.
(a) What is the centripetal force acting on the puck?
(b) How much work is done by the puck in one full circle?
To solve this problem, we'll break it down into two parts:
(a) Finding the centripetal force acting on the puck:
The centripetal force is the force that acts towards the center of the circular path and keeps the puck in its circular motion. In this case, the spring provides the centripetal force.
The centripetal force can be calculated using the formula: F = m * ω² * r, where F is the centripetal force, m is the mass, ω (omega) is the angular velocity, and r is the radius of the circular path.
In this problem, we are given the mass of the puck (m = 4 kg) and the spring constant (k = 100 N/m). The stretch in the spring (Δx) is given as 0.03 meters.
First, let's find the angular velocity (ω):
The angular velocity can be calculated using the equation ω = √(k/m), where k is the spring constant and m is the mass.
ω = √(k/m) = √(100 N/m / 4 kg) = √(25 rad/s²) = 5 rad/s
Now, let's find the radius of the circular path (r):
The radius of the circular path is given as the distance by which the spring is stretched (Δx), which is 0.03 meters.
r = natural length + Δx = 0.07 m + 0.03 m = 0.10 m
Finally, let's calculate the centripetal force (F):
F = m * ω² * r = 4 kg * (5 rad/s)² * 0.10 m = 10 N
Therefore, the centripetal force acting on the puck is 10 N.
(b) Finding the work done by the puck in one full circle:
The work done by a force can be calculated using the formula: work = force * distance * cos(θ).
In this case, the force exerted by the spring is variable, as it depends on the displacement. Since the displacement is perpendicular to the force, θ = 90 degrees, and cos(θ) = 0.
Therefore, the work done by the puck in one full circle is zero.
Hope this helps! Let me know if you have any further questions.