A moving 1.6 kg block collides with a horizontal spring whose spring constant is 267 N/m (see figure). The block compresses the spring a maximum distance of 7.0 cm from its rest position. The coefficient of kinetic friction between the block and the horizontal surface is 0.19. What is the work done by the spring in bringing the block to rest?
(1/2) k x^2 Joules
k = 267 N/m
x = 0.07 m
To find the work done by the spring in bringing the block to rest, we need to consider the different forces acting on the block and calculate the work done by each force.
First, let's calculate the work done by the friction force. The work done by a force is given by the formula:
Work = Force * Distance * cos(angle)
In this case, the force of friction is opposing the motion, so the angle between the force of friction and the displacement is 180 degrees. The distance is the distance over which the friction force acts, which is the compression of the spring. The work done by friction is given by:
Work_friction = Force_friction * Distance * cos(180 degrees)
To calculate the force of friction, we can use the formula:
Force_friction = coefficient of kinetic friction * normal force
The normal force is the force exerted by the surface on the block, which is equal to the weight of the block as there is no vertical acceleration. The weight of the block is given by:
Weight = mass * acceleration due to gravity
Next, let's calculate the work done by the spring. The work done by the spring is given by:
Work_spring = (1/2) * k * x^2
Where k is the spring constant and x is the displacement of the block from its rest position.
Now we have all the information needed to find the work done by the spring in bringing the block to rest.
1. Calculate the weight of the block:
Weight = mass * acceleration due to gravity
2. Calculate the force of friction:
Force_friction = coefficient of kinetic friction * weight
3. Calculate the work done by friction:
Work_friction = Force_friction * Distance * cos(180 degrees)
4. Calculate the work done by the spring:
Work_spring = (1/2) * k * x^2
Add the work done by friction and the work done by the spring to find the total work done by all forces.