# A spring with a constant of #6 (kg)/s^2# is lying on the ground with one end attached to a wall. An object with a mass of #7 kg# and speed of #4 m/s# collides with and compresses the spring until it stops moving. How much will the spring compress?

The compression is

The spring constant is

The kinetic energy of the object is

The mass is

The speed is

This kinetic energy will be stored in the spring as potential energy.

The spring constant is

So,

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To find the compression of the spring, we can use the conservation of mechanical energy. The kinetic energy of the object before the collision will be equal to the potential energy stored in the compressed spring after the collision.

The kinetic energy of the object before the collision is given by: [ KE = \frac{1}{2}mv^2 ]

Where: ( m = 7 ) kg (mass of the object) ( v = 4 ) m/s (speed of the object)

Substituting the values, we get: [ KE = \frac{1}{2}(7 \times 4^2) = 56 , \text{J} ]

The potential energy stored in the compressed spring is given by: [ PE = \frac{1}{2}kx^2 ]

Where: ( k = 6 , \text{kg/s}^2 ) (spring constant) ( x ) is the compression of the spring (in meters)

Equating the kinetic energy before the collision to the potential energy stored in the spring after the collision: [ KE = PE ]

[ 56 = \frac{1}{2} \times 6 \times x^2 ]

Solving for ( x ): [ x = \sqrt{\frac{56 \times 2}{6}} \approx 3.29 , \text{m} ]

So, the spring will compress approximately ( 3.29 ) meters.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- A spring with a constant of #4 (kg)/s^2# is lying on the ground with one end attached to a wall. An object with a mass of #3 kg# and speed of #1 m/s# collides with and compresses the spring until it stops moving. How much will the spring compress?
- An object with a mass of #12 kg# is moving at #9 m/s# over a surface with a kinetic friction coefficient of #1 #. How much power will it take to accelerate the object at #3 m/s^2?
- How much work would it take to push a # 1 kg # weight up a # 4 m # plane that is at an incline of # (5pi) / 12 #?
- A balanced lever has two weights on it, the first with mass #6 kg # and the second with mass #8 kg#. If the first weight is # 4 m# from the fulcrum, how far is the second weight from the fulcrum?
- How much work would it take to push a # 6 kg # weight up a # 2 m # plane that is at an incline of # pi / 4 #?

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