# Math Help - Three problems related to Newton's Laws of Motion

1. ## Three problems related to Newton's Laws of Motion

1. A simple accelerometer consists of a ball of non-zero mass attached to the end of a light, inextensible string as shown in the diagram below. If the angle between the string and the vertical direction is and the constant gravitational field is g, what is the acceleration of the device?

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2. A small block of mass m is placed on top of a frictionless table top. A light, inextensible string is threaded through a hole in the centre of the table and attached to the small block. The other end of the string is attach a large block of mass M which hangs below the table as shown. The small block is then made to move in a circle with radius r from the hole such that the large block remains suspended in equilibrium below. What is the expression for v, the speed of the small block given that there is a constant gravitational field, g.

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3. A ball of mass m is attached to the end of a piece of string. The ball is then given a kick such that it moves in a vertical circle of radius, r. If there is a constant, gravitational field g what is the downwards tension in the string when the ball is at the top of the circle given that at the top of the circle the ball is moving with a speed v?

Any help is greatly appreciated.

2. Hello jpatrie
Originally Posted by jpatrie
1. A simple accelerometer consists of a ball of non-zero mass attached to the end of a light, inextensible string as shown in the diagram below. If the angle between the string and the vertical direction is and the constant gravitational field is g, what is the acceleration of the device?
There is no diagram attached, and we would need to know the velocity of the particle and the length of the string in order to get the acceleration towards the centre.

The acceleration perpendicular to the string (along a tangent to the circle) is given by resolving in this direction, and is simply $g\sin\theta$.

2. A small block of mass m is placed on top of a frictionless table top. A light, inextensible string is threaded through a hole in the centre of the table and attached to the small block. The other end of the string is attach a large block of mass M which hangs below the table as shown. The small block is then made to move in a circle with radius r from the hole such that the large block remains suspended in equilibrium below. What is the expression for v, the speed of the small block given that there is a constant gravitational field, g.
The tension in the string is $Mg$, since the lower mass is in equilibrium. So if we resolve along the string for the upper mass:

$Mg = \frac{mv^2}{r}$

$\Rightarrow v = \sqrt{\frac{Mgr}{m}}$

3. A ball of mass m is attached to the end of a piece of string. The ball is then given a kick such that it moves in a vertical circle of radius, r. If there is a constant, gravitational field g what is the downwards tension in the string when the ball is at the top of the circle given that at the top of the circle the ball is moving with a speed v?
Resolve vertically downwards:

$T + mg = \frac{mv^2}{r}$

$\Rightarrow T = \frac{mv^2}{r}-mg$

3. Here's the picture for question 1.

There is no other information given, such as velocity and the length of the string. sine theta unfortunately was not the answer.

4. Originally Posted by jpatrie
Here's the picture for question 1.

There is no other information given, such as velocity and the length of the string. sine theta unfortunately was not the answer.
grandad did not say the acceleration was $\sin{\theta}$. this is what he stated ...

... and is simply .
...