A body of mass 484 grams was placed
on a smooth plane inclined at an angle 𝜃 to the horizontal, where sin 𝜃 is equal
to three-fifths. Given that a force of 484
gram-weight was acting on the body parallel to the line of greatest slope up the
plane, find the acceleration of motion. Consider the acceleration due to
gravity to be 9.8 meters per second squared.
We will begin by sketching a
diagram to model the scenario. We are told that the body has a
mass of 484 grams. And since there are 1000 gram in a
kilogram, this is equal to 0.484 kilograms. The body exerts a force vertically
downwards equal to its weight, which is equal to 0.484 multiplied by gravity 𝑔. We are told that this is equal to
9.8 meters per second squared. This means that there is a force of
4.7432 newtons acting vertically downwards. There will be a normal reaction
force 𝑅 acting perpendicular to the plane.
We are told that the plane is
inclined at an angle 𝜃 to the horizontal, where sin of 𝜃 is equal to
three-fifths. Since the plane is smooth, there
will be no frictional force. However, there is a force of 484
gram-weight acting up the plane. As this is equal to 0.484
kilogram-weight, we can multiply this by 9.8 meters per second squared, giving us a
force acting up the plane of 4.7432 newtons.
We are now in a position to resolve
parallel to the plane using Newton’s second law 𝐹 equals 𝑚𝑎. The sum of the forces will be equal
to the mass multiplied by the acceleration. We will let this acceleration up
the plane be 𝑎, and we need to calculate the component of the weight force acting
down the plane. We will do this using our knowledge
of right angle trigonometry. The component of the weight force
acting parallel to the plane is 𝑤 sin 𝜃. And whilst it is not required in
this question, the components of the weight force acting perpendicular to the plane
is 𝑤 cos 𝜃.
We have two forces acting parallel
to the plane. And letting the positive direction
be the direction of motion, the sum of our forces is 4.7432 minus 𝑤 sin 𝜃. This is equal to the mass of 0.484
kilograms multiplied by the acceleration 𝑎. Substituting the weight force of
4.7432 newtons and sin 𝜃 equals three-fifths, we have the following equation. Typing the left-hand side into our
calculator gives us 1.89728. This is equal to 0.484𝑎. We can then divide through by 0.484
giving us 𝑎 is equal to 3.92. The acceleration of the body is
3.92 meters per second squared up the plane.