Question

A light spring has unstressed length 15.7 cm. It is described by Hooke's law with spring constant 4.31 N/m. One end of the horizontal spring is held on a fixed vertical axle, and the other end is attached to a puck of mass m that can move without friction over a horizontal surface. The puck is set into motion in a circle with a period of 1.43 s.

(a) Find the extension of the spring x as it depends on m.
x =

Evaluate x for the following masses. (If not possible, enter IMPOSSIBLE.)

(b) m = 0.0700 kg
m?

(c) m = 0.140 kg
m?

(d) m = 0.180 kg
m?

(e) m = 0.230 kg
m?

(f) Describe the pattern of variation of x as it depends on m.

Answer 1

initially, spring is unstretched having length L = 15.7 c = 0.157 m now, when puck is in motion, then, spring would streteched by some length. let that streteched length is x. now, radius of the circular path of the puck = R = L + x time period of the puck around the circular path = T = 1.43s speed of the puck = v = 2piR/T spring force = Fs= kx Here k is the spring constant, k = 4.31 N/m net force on the puck = spring force towards the center Fnet = kx

now, by second law of motion, Fnet = m a (a = centripetal acceleration= v^2/R) m = mass of the puck kx = my^2/R now, lets plug the value of v and R in the above expression we get, KX = m(24R)? kx= 4x² m R² T²R

4л"mR т? х= 45 m (LF) х = 4L + 4mx

ਘਰ - ਸ -x ਘ, ਕਾ ਲੰਨ -

47mL 72k X= I-47²m T2K (extension] Now, this is expression for X Letrs bleeg all values, we geti

4(3:14)?(m) (0.157m) (1.435) ² (4.31 Nlm) 4 (314) ²m 1 (1:435) ? (4:31 H)m) x = -T X= 0.70254m 1-4.4747 m

Now, if m=0.0700kg, then X = 0.70254 (0.070kg) 1-4.4747 (0.0700 kg) - X= 0.0716m - if m=0.140 kg,

- 0.70254 (0.140 kg.) X= 1-4.4747 (0.140 kg x= 0.2633 m if m=0.180 kg, then, 0.70254 (0.18088.) X= 1-4.4747 (0-18 kg)

IX= 0.65m If m=0.230kg. R 0.70254 (0:230 kg.) = -5.537m X= 1-4.4747 (0.230 kg)

This distance is negative, it means it signify for compression in the spring but atually, puck is moving around the circle due to extension of the spring. So, this is not possible.