A few problems (forces, energy, gravitation)

[Destrio]

1. An evacuated box is at rest on a frictionless table. You punch a small hole in one face so that air can enter. How will the box move?

I figure the box will move in the direction opposite that of the hole, the centre of mass will shift in the direction and air will rush into fill the emptyness. Is that thinking correct?

2. A rigid body, starting at rest, rotates about a fixed axis with a constant angular acceleration α. Consider a particle a distance r from the axis. Express (a) the radial acceleration and (b) the tangential acceleration of this particle in terms of α, r and time t.
c) if the resultant acceleration of the particle at some instant makes an angle of 57.0 degrees with the tangential acceleration, through what total angle has the body rotated from t=0 to that instant.

i got radial acceleration = rω^2 = (α^2)(t^2)r
tangential acceleration = αr

for c i made a triangle and got cos(57deg) = cos(.99rad) = (α^2)(t^2)r/αr = αt^2

so i plugged it into the kinematics formula
theta = (1/2)αt^2
theta = (1/2)cos(.99rad)
theta = .27 radians
but that is incorrect

3. Two particles, each with mass m, are fastened to each other and to a rotation axis by two rods, each with length L and mass M. The combination rotates around the rotation axis with angular velocity ω. Obtain an algebraic expression for the rotational inertia of the combination about the axis.

I = m1r1^2 + m2r2^2
I = ML^2 + M(2L)^2
I = 5ML^2
im not sure any other way to solve this

4. A 52.3-kg trunk is pushed 5.95m at constant speed up a 28.0 degree incline by a constant horizontal force. The coefficient of kinetic friction between the trunk and the incline is .19 . Calculate the work done by a) the applied force and b) the force of gravity.

I found the forge of gravity one easily, just mgh

for a) I figured I should make my axes line up with the slope so
Sum of forces in x = Fcos28 - f - mgsin28 = 0
and then solving for F

5. A ball of mass m is projected with speed vi into a barrell of a spring gun of mass M initially at rest on a frictionless surface. The ball sticks in the barrel at the point of maximum compression of the spring. no energy is lost in friction. a) What is the speed of the spring gun after the ball comes to rest in the barrel? b) What fraction of the initial kinetic energy of the ball is lost thorough work done on the spring?

m1v1 = vf(m+M)
Vf = mvi/(M+m)
that's correct

Kfi = (1/2)mvi^2
Kf = (1/2)(m+M)[mvi/(M+m)]^2 = (1/2)m^2vi^2/(m+M)

Kf/Ki = m/(m+M)
the answer says M/(m+M)

6. Two pendulums each of length L are initially situated with one at rest and one being released from height d above the 2nd. Assume the collision is completely inelastic and neglect the mass of the strings and any frictional effects. How high does the centre of mass rise after the collision?

Ui = Uf
mgd = (m1+m2)gh
h = m1d/(m1+m2)

the answer I am told is: d(m1/(m1+m2))^2

7. A boy is seated on the top of a hemispherical mound of ice. He is given a very small push and starts sliding down the ice. Show that he leaves the ice at a point whose height is 2R/3 if the ice is frictionless. (hint: the normal force vanishes as he leaves the ice)

this one I'm really lost on, i don't even know where to begin, do I need to compare his initial potential energy with kinetic energy, or with momentum?

out of my 40 or so final exam review problems these are the ones i couldn't solve, (and 2 more gravitiation ones, but I'll wait for those)

any help is very much appreciated

thanks :)

 

[Destrio]

anyone?

thanks

 

[Destrio]

anyone? please?
im really lost on these

thanks