Center of Mass and Motion of a Wedge-Mass System

[ghostfolk]

Homework Statement

A homogeneous right wedge of mass M, horizontal length L and height H rests on a
horizontal plane, with the side L on the plane.
a) Find the position of the center of mass of the wedge by determining its vertical distance
h from the horizontal plane and its horizontal distance d from the vertex of the right angle
of the wedge.
b) A point mass m is stuck at the top corner of the wedge. Find the new position of
the center of mass of the combined body by calculating the new h and d. (Hint: Using the
decomposition of the center of mass into two parts, as given in class, and already known
results may facilitate the solution.)
The system is initially at rest. At some point in time, the point mass starts sliding down
the sloped side of the wedge, until it reaches its lower corner where it becomes stuck to the
wedge. There is friction between the point mass and the wedge, but there is no friction from
the horizontal plane nor air resistance.
c) Find the final speed of the system mass-wedge.
d) Find the total displacement of the wedge on the horizontal plane at the end of the
motion of the point particle.

I'm stuck on part c and d.

Homework Equations

$F=ma$
$f_k=\mu N$

The Attempt at a Solution

Point mass:
horizontal direction: $ma_p=mgsin\theta-\mu N$
vertical direction: $N-mgcos\theta=0$

Wedge:
horizontal:$F_w=\mu N-Nsin\theta$
vertical: $-Ncos\theta-Mg=0$
Wedge-mass:
$F_{w,p}=(M+m)a$

I'm not entirely sure how to get the acceleration of the wedge-mass system nor the final velocity of the wedge-mass system. Any help is appreciated.[/B]

 

[ehild]

Homework Statement

A homogeneous right wedge of mass M, horizontal length L and height H rests on a
horizontal plane, with the side L on the plane.
a) Find the position of the center of mass of the wedge by determining its vertical distance
h from the horizontal plane and its horizontal distance d from the vertex of the right angle
of the wedge.
b) A point mass m is stuck at the top corner of the wedge. Find the new position of
the center of mass of the combined body by calculating the new h and d. (Hint: Using the
decomposition of the center of mass into two parts, as given in class, and already known
results may facilitate the solution.)
The system is initially at rest. At some point in time, the point mass starts sliding down
the sloped side of the wedge, until it reaches its lower corner where it becomes stuck to the
wedge. There is friction between the point mass and the wedge, but there is no friction from
the horizontal plane nor air resistance.
c) Find the final speed of the system mass-wedge.
d) Find the total displacement of the wedge on the horizontal plane at the end of the
motion of the point particle.

I'm stuck on part c and d.


I'm not entirely sure how to get the acceleration of the wedge-mass system nor the final velocity of the wedge-mass system. Any help is appreciated.

You have the system mass + wedge. Remember the centre of mass theorem: the CM accelerates as if the resultant of the external forces acted at the CM, where all the mass of the system was concentrated.
What are the external forces acting on the system? Do they have any horizontal components?

ehild