A 7 kg block is pushed by an external force against a spring with spring constant 124 N/m until the spring is compressed by 2.1 m from its uncompressed length (x = 0). The block rests on a horizontal plane that has a coefficient of kinetic friction of 0.7 but is NOT attached to the spring.
After all the external forces are removed (so the compressed spring releases the mass) how far D along the plane will the block move before coming to a stop? The acceleration due to gravity is 9.8 m/s2 .
Answer in units of m
PE=W(fr)
kx²/2=μmgd
d= kx²/2μmg=
=124•2.1²/2•0.7•7•9.8 =5.7 m
Sorry but that was wrong
To determine the distance D along the plane that the block will move before coming to a stop, we need to consider the forces acting on the block.
The initial external force applied to the block compresses the spring by 2.1 m. The work done by this external force is equal to the potential energy stored in the compressed spring:
Work = (1/2) * k * x^2
where k is the spring constant and x is the compression distance.
In this case, the work done by the external force is given by:
Work = (1/2) * 124 N/m * (2.1 m)^2
Next, we need to consider the friction force acting on the block. The friction force can be calculated using the coefficient of kinetic friction and the normal force.
The normal force is equal to the weight of the block, which can be calculated by multiplying the mass of the block (7 kg) by the acceleration due to gravity (9.8 m/s^2).
Normal force = mass * acceleration due to gravity
Normal force = 7 kg * 9.8 m/s^2
The friction force can then be calculated using the coefficient of kinetic friction:
Friction force = coefficient of kinetic friction * normal force
Friction force = 0.7 * (7 kg * 9.8 m/s^2)
Now, we can calculate the net work done on the block. Since the block eventually comes to a stop, the net work done is equal to the work done by the external force minus the work done by friction:
Net Work = Work - Friction force
Finally, we can use the work-energy principle to calculate the distance D along the plane. The net work is equal to the change in kinetic energy, which is then related to the distance D:
Net Work = (1/2) * mass * (final velocity)^2
Net Work = (1/2) * 7 kg * (final velocity)^2
Setting the net work equal to the above expression and solving for the final velocity, we find:
(final velocity)^2 = (2 * (Work - Friction force)) / mass
final velocity = sqrt((2 * (Work - Friction force)) / mass)
The distance D can then be calculated using the equation:
D = (1/2) * acceleration due to gravity * (time)^2
Since the final velocity is zero when the block comes to a stop, we can rearrange the equation for D as follows:
D = (initial velocity) * (time) + (1/2) * acceleration due to gravity * (time)^2
In this case, the initial velocity is zero, so we can simplify the equation to:
D = (1/2) * acceleration due to gravity * (time)^2
Now, we can plug in the calculated values and solve for D.