Incline plane + Spring?

Question

A mass of 8.60 kg rests on a smooth surface inclined 37.0(deg) above the horizontal. It is kept from sliding down the plane by a spring attached to the wall. the spring is aligned with the plane and has a spring constant of 125 N/m. How much does the spring stretch?

Answer 1

FBD of the mass

Sum of the forces along the plane = 0 = F(spring) - m*g*sin(theta)

Fspring = m*g*sin(theta)

Force from a spring is defined as F = k*x.

k*x = m*g*sin(theta)

x = m*g*sin(theta)/k

presto!

I got x = 0.41 m

Answer 2

Since the mass is resting, we know that it is not accelerating. Therefore the forces acting parallel to the surface of the plane must be in balance. There are only two such forces we need to consider. The first is the weight of the mass, which has a component = mgsin(37.0) along the plane.

mgsin(37.0)= (8.60)(9.80)sin(37.0)=50.7 Newtons.

This force must be offset by the spring. If the spring stretches S meters, it exerts a force of 125S Newtons. So we have :

125S=50.7
S=50.7/125 = 0.406 meters

The spring stretches 40.6 cm

Answer 3

The force of gravity on the mass is acting vertically downwards. You need to resolve it into two components: one along the direction of the inclined plane (which will stretch the spring), and one at right-angles to the inclined plane (which will not).

Represent the weight of the mass (= 8.6 * g) by a vector pointing down. This is the hypotenuse of a right-angled triangle. The other two sides -- one of which is parallel to the direction of the inclined plane, i.e. 37 degrees below the horizontal -- are the vectors representing the forces parallel and perpendicular to the plane. We can calculate (by trigonometry) or measure (by scale drawing) the length of the side parallel to the plane, and this gives us the component of the force acting in that direction. Then we just need to work out how much the spring stretches with this amount of force, if it takes 125N to stretch the spring by 1m.

Force component parallel to plane = F = 8.6 * g * sin(37 deg.)
Amount of stretch = F / 125.