Figure shows a smooth track, a part of which is a circle of radius 10 cm. A block of mass 3 kg is pushed against a spring of spring constant 100 N/m fixed at the left end and is then released. If the initial compression of the spring is 10x cm, find the value of x so that the block pushes the track with a force of 30 N when it reaches point P, where the radius of the track is horizontal.
Physics · Mixed Chapter
JEE Mains

Question

Figure shows a smooth track, a part of which is a circle of radius 10 cm. A block of mass 3 kg is pushed against a spring of spring constant 100 N/m fixed at the left end and is then released. If the initial compression of the spring is 10x cm, find the value of x so that the block pushes the track with a force of 30 N when it reaches point P, where the radius of the track is horizontal.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1: The force at P is radial since the track radius is horizontal. Therefore N=mv²/R gives v²=(30×0.10)/3=1 m²/s². Step 2: Point P is R=0.10 m above the bottom of the curved track. By energy conservation, initial spring energy supplies both kinetic and gravitational energies: (1/2)k(0.10x)²=(1/2)mv²+mgR. Step 3: (1/2)(100)(0.01x²)=(1/2)(3)(1)+(3)(10)(0.10) ⇒0.5x²=4.5. Step 4: x=3. Answer: 3.