An object of mass m can slide without friction on a tabletop. A horizontal string tied to the object passes through a frictionless hole in the table and is connected to a second, hanging mass, M. If m is put into motion so that it moves in a circular path of radius r, find

a) the tension in the string.
b) the speed of m.

To find the tension in the string and the speed of m, we can use the principle of centripetal force.

a) The tension in the string provides the centripetal force that keeps the object m moving in a circular path. According to Newton's second law of motion, the centripetal force is given by the equation:

Tension = Centripetal Force

Since the object is sliding without friction on the tabletop, the only force acting on it is gravity pulling it down. Therefore, the centripetal force is provided by the weight of the hanging mass M.

Centripetal Force = Weight of M

The weight of an object is given by the formula:

Weight = Mass * Acceleration Due to Gravity

So we can now write:

Tension = Weight of M

Tension = M * g

where g is the acceleration due to gravity.

b) The speed of m can be found using the centripetal force equation:

Centripetal Force = (Mass * Speed^2) / Radius

Since the tension in the string provides the centripetal force, we can write:

Tension = (m * v^2) / r

where v is the speed of m.

Now we have a system of equations:

Tension = M * g
Tension = (m * v^2) / r

We can solve these equations simultaneously to find both the tension and the speed.

To eliminate the tension, we can set the two equations equal to each other:

M * g = (m * v^2) / r

Rearranging the equation, we get:

v^2 = (M * g * r) / m

Finally, we take the square root of both sides to find the speed of m:

v = √((M * g * r) / m)

Note that the negative sign is ignored because we are only interested in the magnitude of the speed.

To find the tension in the string and the speed of m, we can apply some principles from physics.

a) Tension in the string:
The tension in the string can be found by considering the forces acting on the hanging mass M. There are two forces acting on M: its weight (mg) and the tension in the string.

The weight of M (mg) can be equated to the centripetal force acting on M, which is provided by the tension in the string.

Centripetal force (Fc) = mg

Since the circular motion of m is caused by the tension in the string, we can equate the centripetal force acting on m with the tension in the string.

Centripetal force (Fc) = Tension in string

Therefore, the tension in the string is equal to the weight of M:
Tension in string = mg

b) Speed of m:
To find the speed of m, we can make use of the centripetal force acting on m.

The centripetal force (Fc) acting on m is provided by the tension in the string.

Centripetal force (Fc) = Tension in string

The centripetal force can also be expressed as the product of mass (m) and acceleration (a), where a is the centripetal acceleration.

Centripetal force (Fc) = m * a

The centripetal acceleration (a) can be calculated using the formula:
a = v² / r
Where v is the speed of m and r is the radius of the circular path.

Equating the centripetal force expressions, we can write:
m * a = Tension in string

Substituting the values:
m * (v² / r) = mg

Simplifying the equation, we get:
v² = g * r

Taking the square root of both sides, we find:
v = √(g * r)

So, the speed of m is given by the square root of the product of acceleration due to gravity (g) and the radius (r) of the circular path.

In summary:
a) The tension in the string is equal to the weight of the hanging mass: Tension in string = mg.
b) The speed of m is given by: v = √(g * r).