# Find the maximum velocity of the block

• PhilCam
In summary, a small cart attached to a horizontal spring with a mass of 0.8 kg and a spring constant of 600 N/m is released from rest after being stretched by 0.05 m. The cart oscillates back and forth with negligible friction and ignoring the mass of the spring. The angular frequency of the oscillations can be found using the equation ω=√(k/m), where k is the spring constant and m is the mass of the cart. The maximum velocity of the cart can be found using the formula v=√(kx^2/m), where x is the maximum displacement of the cart. At the point where the acceleration is zero, the total energy of the system is equal to the
PhilCam

## Homework Statement

A small cart of mass .8 kg is attached to one end of a horizontal spring as shown in the figure. The spring constant is 600 N/m. When the spring is stretched by .05 m and the car is released from rest, the car oscillates back and forth. The friction is negligible. Ignore the mass of the spring.
A) Find the angular frequency of oscillations in HZ.
B) Find the maximum velocity of the block.
C) Find the total energy of the system at the point where acceleration is zero.

## Homework Equations

Angular frequency = 2(3.14)F

## The Attempt at a Solution

Pretty lost, if anyone could guide me to what formulas to use or what way to attack this problem I would be very grateful. Thanks

A) there is a much simpler equation for angular frequency using things you're given. they're found by solving SHO differential equations, did you do this in class?

B) the maximum velocity will occur when the acceleration equals...?

C) there's two types of energy involved. what are they?

A) I have not found any other equation in my notes and to be honest, I'm not sure what "SHO" is.

B) When acceleration is zero, velocity will be maximum.

All the formulas I have for finding velocity include acceleration and by setting it to zero, it would be unsolvable.

I have formulas like F=kx but nothing that incorporates spring constant into velocity.

C) The two forces are potential and kinetic energy.
PE=1/2kx^2
PE= (.5)(600)(.05m)^2
PE= .75
is .75 Joules?

SHO stands for simple harmonic motion. Have you seen the equation T=2pi*sqrt(m/k)?

Sorry, SHO is Simple Harmonic Oscillation. $$\omega=\sqrt{\frac{k}{m}}$$

What formulas for velocity do you have?

I think (but am not sure) that when the acceleration is 0, the spring would be at equilibrium (not stretched or compressed) therefore making the equation you used for energy equal to 0. Since they ask for max velocity (when acceleration is 0) and THEN ask for the energy when acceleration is equal to zero, you could use the normal kinetic energy equation, KE=(1/2)mv^2, using the maximum velocity you find in B.

Could anyone else confirm this?

Confirmed. Alternatively, you could say that the kinetic energy is equal to the max potential energy, which is 0.5*k*x^2

Thanks for the help guys.

I have figured out A and C but am still unsure on part B.

would conservation of energy work?
we now know that (1/2)mv^2=(1/2)kx^2, so try solving for v.

Worked like a charm. thanks again.

No problem!

## 1. What is the maximum velocity of the block?

The maximum velocity of a block can be determined by calculating the acceleration of the block and using the equation V=at, where V is the velocity, a is the acceleration, and t is the time. The highest value obtained from this equation is the maximum velocity.

## 2. How is the maximum velocity related to the block's acceleration?

The maximum velocity and acceleration of a block are directly proportional. This means that as the acceleration of the block increases, so does the maximum velocity. However, the maximum velocity can also be influenced by other factors such as friction and air resistance.

## 3. What factors can affect the maximum velocity of a block?

The maximum velocity of a block can be affected by various factors such as the surface it is moving on, the force applied to it, and any external forces such as air resistance or friction. These factors can either increase or decrease the maximum velocity of the block.

## 4. Can the maximum velocity of a block ever be exceeded?

No, the maximum velocity of a block is the highest speed it can reach in a given environment. It cannot be exceeded, even with a greater force or acceleration. This is because the maximum velocity is limited by the factors mentioned earlier, such as friction and air resistance.

## 5. How can the maximum velocity of a block be useful in real-life situations?

The concept of maximum velocity is important in many real-life situations, especially in engineering and physics. It helps engineers determine the maximum speed a vehicle can reach, or the maximum load a structure can hold. In physics, it is used to study the motion of objects and understand their limits and capabilities.

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