A body of mass four kilograms is moving down an inclined smooth plane with angle 30 degrees under its own weight. If the body starts its motion from rest, what is the change in potential energy of the body, in joules, two seconds after it starts moving? Let 𝑔 equal 9.8 meters per second squared.
In this question, we have a scenario where gravitational potential energy is being converted into kinetic energy. Since the body is moving down a smooth plane, we have no friction and no dissipative forces. Therefore, by the principle of the conservation of energy, the change in gravitational potential energy, 𝐸 𝑔, will be equal and opposite to the change in kinetic energy of the body, 𝐸 𝑘. We therefore need to find the change in kinetic energy of the body after two seconds.
Recall that the kinetic energy of a body of mass 𝑚 moving with a speed 𝑣 is given by 𝐸 𝑘 equals one-half times 𝑚𝑣 squared. If the body starts with an initial speed of 𝑣 naught and finishes with a final speed of 𝑣 one, then the change in kinetic energy Δ𝐸 𝑘 will be equal to half 𝑚𝑣 one squared minus half 𝑚𝑣 naught squared. In this case, the body is starting from rest. Therefore, 𝑣 naught is equal to zero, so this whole term will be zero. We have the mass 𝑚 of the object equal to four kilograms. Therefore, all we need to do is find the speed of the body two seconds after it starts moving.
Let’s consider a diagram of the scenario. We have a body of mass four kilograms moving down an inclined smooth plane at an angle of 30 degrees to the horizontal. The weight of the body acts vertically downwards and is equal to the body’s mass, four, multiplied by the acceleration due to gravity, 𝑔. The only other force on this body is a reaction force 𝑅 which acts perpendicularly to the plane. Since the body remains on the plane, by Newton’s third law, the body also exerts a force 𝑅 perpendicularly onto the plane, which is given by the parallel component of four 𝑔. This component 𝐹 of the body’s weight is parallel to the plane and is what causes the object to move by Newton’s second law.
By resolving triangles, we can deduce that this angle here in this right triangle is 30 degrees. The component 𝐹 is on the opposite side of this triangle. Therefore, 𝐹 is equal to the hypotenuse, four 𝑔, multiplied by sin of 30 degrees. sin of 30 degrees is one-half. Therefore, this simplifies to two 𝑔. Now, by Newton’s second law, force is equal to mass times acceleration. So 𝑚𝑎 equals two 𝑔. Therefore, 𝑎 is equal to two 𝑔 over 𝑚. And 𝑚 is equal to four kilograms, so 𝑎 is two 𝑔 over four, which is equal to 𝑔 over two.
We now need to use the kinematics equations, also known as the SUVAT equations, to find the velocity of the object after two seconds. We want to find the final velocity. And we have the initial velocity, zero. And we have the acceleration, 𝑔 over two. And we have the time, two seconds. Therefore, the kinematics equation we need is 𝑣 equals 𝑢 plus 𝑎𝑡, where 𝑢 is the initial velocity, 𝑎 is the acceleration, and 𝑡 is the time. Since the body starts from rest, 𝑢 is equal to zero. We already have 𝑎 equal to 𝑔 over two and that the time 𝑡 is two seconds. Therefore, the velocity after two seconds is just equal to 𝑔 over two times two, which is equal to 𝑔, which is equal to 9.8 meters per second. Note that the units here are not the same as 𝑔 since we multiplied by time, which has dimension seconds.
We now have everything we need to find the change in kinetic energy given by half 𝑚𝑣 one squared. So this is equal to one-half times four times 9.8 squared, which comes to 192.08 joules. By the principle of the conservation of energy, this is equal in magnitude to the change in gravitational potential energy. Therefore, the change in gravitational potential energy is also equal to 192.08 joules.