When an object is moving in a circular path, the velocity at the topmost point of the path can be analyzed in the context of uniform circular motion. Here's a breakdown:
1. **Definition of Velocity in Circular Motion**: Velocity in circular motion is always tangential to the path. This means the speed is constant if the motion is uniform, but the direction of velocity changes continuously.
2. **Topmost Point in Circular Motion**: At the topmost point of a vertical circular path, gravity acts downward while the centripetal force needed to keep the object in circular motion also acts downward. The net force acting on the object provides the centripetal force required for circular motion.
3. **Expression for Velocity**: The centripetal force \(F_c\) required to keep the object moving in a circle of radius \(r\) with velocity \(v\) is given by:
\[
F_c = \frac{mv^2}{r}
\]
At the topmost point, the gravitational force \(mg\) helps provide the centripetal force, so:
mg + N = \frac{mv^2}{r}
where \(N\) is the normal force at the topmost point. For an object just moving in the circle (minimal normal force), \(N\) can be approximated as zero:
mg = \frac{mv^2}{r}
Solving for \(v\):
v = \sqrt{gr}
In summary, at the topmost point of a vertical circular path, the velocity \(v\) can be found using \(v = \sqrt{gr}\), where \(g\) is the acceleration due to gravity and \(r\) is the radius of the circle.
Question : If a body moves with a constant speed in a circle:
Option 1: no work is done on it
Option 2: no force acts on it
Option 3: no acceleration is produced
Option 4: its velocity remains constant
Question : In 1851, who founded the science of hydrodynamics with his law of viscosity describing the velocity of a small sphere through a viscous fluid?
Option 1: Daniel Bernoulli
Option 2: George Gabriel Stokes
Option 3: Evangelista Torricelli
Option 4: Heinrich Gustav Magnus
Question : Comprehension:
Read the following passage and answer the questions given after it.
The Indian Space Research Organisation (ISRO) announced another successful mission. ISRO chairman S. Somanath said that three small satellites successfully separated and were injected into orbit. ISRO launched three small satellites powered by its SSLV-D2 launch vehicle from Satish Dhawan Space Centre at Sriharikota on February 10, 2023, at 09:18 AM IST. The launch vehicle was carrying EOS-07, Janus-1 & AzaadiSAT-2 satellites and aimed to inject them into a 450 km circular orbit. The launch took place at the first launch pad at SDSC SHAR, Sriharikota. The SSLV-D2 had to take a 15-minute flight to inject EOS-07, Janus-1, and AzaadiSAT-2 satellites into a 450 km circular lower orbit. “It is configured with three solid propulsion stages and a velocity terminal module. It is a 34 m tall, 2 m diameter vehicle having a lift-off mass of 120 t,” said ISRO. The EOS-07 satellite weighs 156.3 kg and is made by ISRO.
About 13 minutes into its flight, the SSLV rocket ejected EOS-07 and soon after that the other two satellites Janus-1 and AzaadiSAT-2 were ejected — all at an altitude of 450 km, said ISRO.
With the new rocket in its portfolio, ISRO will have three rockets — Polar Satellite Launch Vehicle (PSLV) and its variants (cost about Rs 200 crore), Geosynchronous Satellite Launch Vehicle (GSLV-MkII cost about Rs 272 crore and LVM3 Rs 434 crore) and SSLV (Development cost of three rockets about Rs 56 crore each) and production cost may go down later.
“New experiments include mm-Wave Humidity Sounder and Spectrum Monitoring Payload. Janus-1, a 10.2 kg satellite belongs to ANTARIS, USA. An 8.7 kg satellite AzaadiSAT-2 is a combined effort of about 750 girl students across India guided by Space Kidz India, Chennai,” it added.
SSLV caters to the launch of up to 500 kg satellites to Low Earth Orbits on a ‘launch-on-demand’ basis. “It provides low-cost access to Space, offers low turn-around time and flexibility in accommodating multiple satellites, and demands minimal launch infrastructure,” as per a statement by ISRO.
Question:
After reading the passage it can be inferred that it is
Option 1: a government release
Option 2: a news item
Option 3: a magazine article
Option 4: an editorial
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