a) On the axes provided below, draw graphs of the ball’s position y(t), velocity

Question

a) On the axes provided below, draw graphs of the ball’s position y(t), velocity vy(t), and acceleration ay(t) as functions of time t. Synchronize the t axes of your graphs, and show 2 complete periods of the motion. Mark the maximum and minimum values ±A, ±vmax, and±amax on the vertical axes.

How do the frequencies of these oscillations compare?

Are these oscillations in-phase with each other or are they phase-shifted? If they are phase- shifted, by what fraction of a cycle relative to y(t) is v(t) phase-shifted? Same question about the phase shift of a(t) with respect to y(t).

b) Using Newton’s 2nd Law and your previous results, write an expression for the net forceFnet(t) acting on the ball as a function of time t.

What are the maximum (it will be positive, i.e. pointing up) and minimum (it will be negative, i.e. pointing down) values of Fnet(t)?

c) Draw a carefully labeled free-body diagram for the ball, assuming that it remains in contact the hand. Write Newton’s 2nd Law for the ball, and use it to derive an expression for the normal contact force N(t) due to the hand as a function of time t.

What are the maximum and minimum values of N(t)?

d) On the axes on the provided below, draw graphs of the net force Fnet(t) acting on the ball and the normal contact force N(t) due to the hand as functions of time t. Mark values for the maximum and minimum values of these forces on the vertical axes.

2.2 Problem 2 A ball of mass m sits on a hand that that undergoes vertical sinusoidal oscillatory motion with amplitude A and frequency f. The hand's vertical position as a function of time t is y(t) = Asin (2?ft). Our goals are to examine the kinematics and dynamics of this motion and to determine the conditions on A and f for which the ball remains in contact with the hand throughout the hand's oscillation cycle. Figure 2: Ball on a hand

Explanation / Answer

due to discrepancy in chegg upload module, the image cannot be uploaded, hence the equations are shown for illustration

a. lets assume the ball is at origin at t = 0, and starts moving with initial speed vo in the upward direction

hence, then for the sinusoildal motion the equations of motion of the ball are

y(t) = A*sin(wt) [ where A is amplitude of motion of the ball, w is the angular frequency of osscilation w = 2*pi/T , where Y is time period of oscilation ]

similiarly

v(t) = Aw*cos(wt) = dy/dt

a(t) = -Aw^2*sin(wt) = -w^2*y

hecne the grpah of v(t) is phase shifter by 90 deg to the left wrt y(t)

and a(t) is further phase shifter to the left by 90 deg wrt v(t)

b. fom newtons second law

net force

F = m*a = -mAw^2*sin(wt)

c. the normal force will be acting upwards on the ball

the weight will be acting downwards

hence

N(t) - mg = ma = -mAw^2*sin(wt)

N(t) = m(g - Aw^2*sin(wt))

hence N(t) max = m(g + Aw^2)

N(t) min = m(g - Aw^2)

d. hence for the ball to remain in contact with the hand always

N(t) min > 0

hence

m(g - Aw^2) > 0

Aw^2 < g

hence

4*pi^2*A/T^2 < g

for initial amplitude A

T^2 > 4*pi^2*A/g

T > 2*sqrt(A)

hence

this condition must be met for the ime period of the osscilation and the amplitudeof the osscilation corelation for he ball never to lea ve contact with the hand

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