A mass M of 2.95 kg is attached to the end of a string whose length is 0.420 m. The mass slides without friction on a horizontal surface as indicated in the diagram.

If the string can withstand a maximum tension of 127.5 N, what is the maximum tangential speed the ball can have before the cord breaks?

The mass, with a new string, is whirled in a vertical circle of the same radius about a fixed point. Find the magnitude of the tension when the mass is at the top if its speed at the top is 7.11 m/s.

i have no idea

To find the maximum tangential speed before the cord breaks, we need to consider the tension in the string at that speed.

The tension in the string is related to the centripetal force acting on the mass. In this case, the centripetal force is provided by the tension in the string.

According to Newton's second law, the centripetal force is given by the equation:

F = (m * v^2) / r

Where:
F is the centripetal force
m is the mass of the object
v is the tangential speed
r is the radius of the circular motion

In this case, the centripetal force is equal to the maximum tension the string can withstand, which is 127.5 N. The mass is given as 2.95 kg, and the radius is the length of the string, which is 0.420 m.

Plugging in these values into the equation, we can solve for the maximum tangential speed:

127.5 N = (2.95 kg * v^2) / 0.420 m

Simplifying the equation:

v^2 = (127.5 N * 0.420 m) / 2.95 kg

v^2 = 18.989 N*m / 2.95 kg

v^2 = 6.442 N*m/kg

Taking the square root of both sides, we find:

v = √(6.442 N*m/kg)

Calculating the value of v:

v = 2.54 m/s

Therefore, the maximum tangential speed the ball can have before the cord breaks is 2.54 m/s.

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To find the tension in the string when the mass is at the top of the vertical circle, we need to consider the forces acting on the mass at that point.

At the top of the circle, the tension in the string provides the necessary centripetal force to keep the mass moving in a circular path. However, there is also the force of gravity acting on the mass.

The equation for the tension at the top is:

T = m * g + (m * v^2) / r

Where:
T is the tension in the string
m is the mass of the object
g is the acceleration due to gravity
v is the speed of the object at the top
r is the radius of the circular motion

In this case, the mass is still 2.95 kg and the radius is the same as before, 0.420 m. The speed at the top is given as 7.11 m/s, and the acceleration due to gravity is 9.8 m/s^2.

Plugging in these values into the equation, we can solve for the tension:

T = (2.95 kg * 9.8 m/s^2) + (2.95 kg * (7.11 m/s)^2) / 0.420 m

Simplifying the equation:

T = 28.91 N + (2.95 kg * 50.5921 m^2/s^2) / 0.420 m

T = 28.91 N + 142.118 N

T = 171.028 N

Therefore, the magnitude of the tension when the mass is at the top of its vertical circle with a speed of 7.11 m/s is 171.028 N.