Question

An experiment is set up as follows:
A mass m = 9.4kg is sent down a frictionless ramp with an initial velocity of 2.5m/s. The ramp is 87cm
long and has an angle of 36°above the horizontal. At the bottom of the ramp is a horizontal surface with a
kinetic friction coefficient of μk = 0.27. At the far side of the horizontal surface, 48cm away, is a spring with
k-constant ks = 3,413.7N
m that will be compressed as the mass collides with the spring. The experiment
ends as the spring is fully compressed and the mass is at rest.
Note: The distance that the spring is compressed is in addition to the 48cm.
7) Find the initial energy of the mass. (10pts)
8) How far will the spring compress if there is no surface friction under the spring? (10pts)
9) How far will the spring compress if the surface friction continues under the spring? (20pts)

Answer 1

(a) The initial energy of the mass is 76.485 J.

(b) The compression of the spring in the absence of friction is 21.2 cm.

(c)  The compression of the spring in the presence of friction is 19.4 cm.

Initial energy of the mass

The initial energy of the mass is determined as follows;

E = K.E + P.E

E = ¹/₂mv² + mgh

where;

• h is the height of the ramp

E = ¹/₂mv²  + mg x Lsinθ

P.E = ¹/₂(9.4)(2.5)² +  (9.4)(9.8)(0.87)(sin36)

P.E = 76.485 J

Compression of the spring when there is no surface tension

The compression of the spring in the absence of friction is calculated as follows;

Ux = E

¹/₂kx² = 76.485

kx² = 2(76.485)

x² = (2 x 76.485)/k

x = √(2 x 76.485)/k

x = √(2 x 76.485 / 3413.7)

x = 0.212 m

x = 21.2 cm

Compression of the spring in presence of friction

The compression of the spring in the presence of friction is determined by applying the principle of conservation of energy.

E - Fd = Ux

E - μmgd = ¹/₂kx²

76.11 - (0.27 x 9.4 x 9.8 x 0.48) = ¹/₂(3413)x²

64.17 = 1706.5x²

x² = 0.0376

x = √0.0376

x = 0.194

x = 19.4 cm

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